Deformation geometry of cellulose fibril arrays constraining the stretching and growth of plant cell walls

There are several ways in which the nanoscale array of cellulose fibrils in one layer of a plant cell wall can rearrange to permit the cell wall to e xpand under stress. To be uniform enough for analysis, a cell-wall layer may consist of a single lamella of microfibrils in a primary cell -wall or the dominant S2 layer in a wood cell wall. The stress may be external uniaxial tension or, in the case of a primary-wall layer, the biaxial turgor pressure that drives growth. Under uniaxial tension, most cell-wall domains narrow as they elongate. During growth, cell diameter and hence the width of each cell-wall domain usually stay constant or increase in a controlled way. The balance of nanoscale deformation modes generates these cell-scale changes in dimensions. Here, seven such deformation modes are identified and their geometry is described for rectangular cell-wall domains with constant initial microfibril angle, a simplified model for one facet of a cell -wall. These deformation modes are fibril rotation, regular shear, i nterdigitated sliding, fibril stretching, widened fibril spacing and the formation and straightening of waves. The distinction between regular shear and interdigitated sliding is introduced to capture a continuous range of sliding modes at fibril interfaces, with varying proportions of sliding in opposite directions and varying outcomes for cell shape. An unforeseen finding was that the functions representing some of the deformation modes apply only to a rectangular domain of moderate aspect ratio, representing one facet of the wall of a cell of moderate length. For longer or shorter domains, different f unctions apply and the relative elongation of the domain diminishes as the aspect ratio rises. It is noticeable that cells elongating in vivo often remain within that moderate range. The scale-independent geometric descriptions presented here are a prerequisite for predicting how several modes of nanoscale deformation interact to create the observed changes in the mesoscale dimensions of the cell and thus to allow growth to shape the emergence of multicellular plant form. Key words: microfibrils, orientation, shear, sliding, spacing, tension, elongation, growth .CC-BY-NC-ND 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted December 25, 2025. ; https://doi.org/10.64898/2025.12.23.696159doi: bioRxiv preprint 2

Background In multicellular plants, e very cell is fixed to the next through their cell walls. The primary cell wall, through its unique capability to stretch without losing strength (Cosgrove 2024b), controls how turgor pressure drives anisotropic growth of each cell (Zhang, Ramakanth, Long 2024) in concert with the growth of its neighbours (Kelly-Bellow et al. 2023). Cell-wall stretching and synthesis of new wall material are synchronised (Muller, Drevensek, Boudaoud 2025) . From these properties of primary cell walls emerge the growth and diverse forms of living plants (Coen and Cosgrove 2023) . The secondary walls of cells that have ceased to grow provide strength, reinforcing plants against stresses like wind and the negative pressures entailed by xylem translocation (Jarvis 2024). Both primary and secondary cell walls are assembled from arrays of cellulose fibrils embedded in non-cellulosic polymers (Cosgrove 2024b). There are recent reviews of the polymer structures (Salmén 2022; Cosgrove 2024b; Delmer et al. 2024) and new insights are still emerging , e.g. (Xiao et al. 2025; Schoenaers et al. 2024; Cresswell et al. 2025) . A key factor in growth and strength is the microfibril angle ϑ between the mean fibril orientation and the cell axis, which results initially from the direction in which the cellulose synthase complexes travel when laying down microfibrils (Coen and Cosgrove 2023) . The microfibril angle depends on developmental factors (Coen and Cosgrove 2023; Schneider et al. 2021) and, it is proposed, by feedback from forces within the cell wall (Lan et al. 2025; Belteton et al. 2021; Muller, Drevensek, Boudaoud 2025). Structural changes observed under mechanical tension have been described by, e.g. (Thomas et al. 2021; Zhang, Yu, Cosgrove 2025) . It is now considered that direct cellulose -cellulose interactions between fibrils are important in the mechanical properties of both primary (Zhang, Yu, Cosgrove 2025) and secondary (Thomas et al. 2021) cell walls, but these interactions are not continuous along each fibril. Particularly in primary cell walls, cellulose surfaces adhere only in short segments (Cosgrove 2024b) , here called junction zones (Morris et al. 1982) , forming an anastomosing network. At the nanometre scale there are various ways in which a small group of fibrils could rearrange to make a local contribution to the expansion of the whole cell wall under stress. It is presumably at this local level in primary cell walls that growth is controlled, although we do not understand the polymer structures and interactions well enough to be sure of the detailed molecular mechanisms responsible (Pfaff, Wagner, Cosgrove 2024) and hence the relative energy demands of the possible modes of fibril rearrangement. While elongation growth has received most attention, accompanying lateral expansion also shapes plant organs. Similar questions surround the strength of wood as an engineering material (Thomas et al. 2021) . In a landmark paper on the stretching of wood under tension, Keckes et al. (2003). showed how two nanoscale deformation modes, shear between fibrils and rotation of the fibril array, co -operate geometrically to allow elongation. Since .CC-BY-NC-ND 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted December 25, 2025. ; https://doi.org/10.64898/2025.12.23.696159doi: bioRxiv preprint 3 then, several other local deformation modes of fibril arrays have been imaged, or inferred from diffraction data, or predicted from coarse-grained simulations; in wood, primary cell walls or both. These deformation modes include fibril stretching (Thomas et al. 2021), widening of the spaces between fibrils (Marga et al. 2005) , forms of sliding different (Yu et al. 2024; Thomas et al. 2021) from the regular shear assumed by Keckes et al . (2003), and .the formation or straightening of waves (Yu et al. 2024) . The rules de fining how all these modes of deformation interact and contribute to cell- wall expansion have not yet been established, although useful insights have come from coarse -grained simulations (Zhang et al. 2021b).

The present paper draws on concepts originating in both wood and primary cell walls to arrive at an analytical description of the geometry of each of the known local deformation modes of fibril arrays. Restricting the analysis to geometry allows it to be scale -independent. Thus, nanoscale functions can be scaled up to cell -scale inferences, and it is not a problem that, whereas 3 nm microfibrils are the elementary fibrillar units of primar y walls (Cosgrove 2024b), macrofibrils – bundles of microfibrils – form topologically similar anastomosing arrays in secondary walls (Thomas et al. 2021) . In primary cell -walls, each mode of nanoscale deformation is also a point at which there is potential to control the magnitude and direction of growth that

(Cosgrove 2024b) . In wood, each mode of deformation is a link between nanostructure and macroscopic resistance to stress (Thomas et al. 2021).

Working Assumptions This study deals with the geometry of deformation. It concerns only strain, not stress. It includes some qualitative

of the amount of rearrangement needed for each deformation mode, but there is no quantitative consideration of the forces or energies required. It is assumed that we are looking at one cell- wall layer in which the fibrils are homogeneous with respect to overall orientation. This could approximate to the fibril geometry in a single lamella of a primary wall, one microfibril (<10 nm) thick, or in the S2 lamella of a secondary cell wall, some hundreds of microfibril layers in thickness (ca. 1 m). Interactions between layers with differing microfibril geometry are not systematically considered here. Domains of deformation The functions describing elongation, change in width and twist differed with the shape of the cell -wall domain under consideration. The rectangular domain assumed by Keckes et al. (2003) corresponded to the cell-wall of a softwood tracheid cut open and laid out flat. However, if the domain is to be a homogeneous rectangle, it seems sensible to consider a single wall facet rather than flattening out all the longitudinal walls. In softwoods the tangential walls are uniform but the radial walls contain regions (pit fields) with different cellulose orientation (Sedighi-Gilani, Sunderland, Navi 2005) . In growing tissues the microfibril orientation differs at the cell edges (Lee et al. 2023) . In many cell types, each wall facet is a rectangle with varying length/width ratio (aspect ratio), although there are also cells with much more complex shapes (van .CC-BY-NC-ND 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted December 25, 2025. ; https://doi.org/10.64898/2025.12.23.696159doi: bioRxiv preprint 4 Spoordonk, Schneider, Sampathkumar 2023). The cell edge domains (Lee et al. 2023) can be examined separately as distinct, narrow rectangles. For rectangular domains, some of the dimensional functions turned out to be discontinuous with respect to the aspect ratio of the rectangle and the microfibril angle θ. Discontinuous dimensional functions are also a feature of other shapes, such as a stretched hexagon (Yu et al. 2024), that could correspond to one wall of a plant cell. If the aim is to examine the deformation of a small domain within a spatially non-uniform cell wall, e.g. an epidermal pavement cell (Elsner, Kwiatkowska, Borowska-Wykret 2025), a cell from which a hair is initiating (Liu et al. 2022) , or a pit -field region in a radial wood cell -wall (Sedighi-Gilani, Sunderland, Navi 2005) , it is more convenient to assume a circular domain deforming to an ellipse. With circular geometry discontinuous functions are avoided, together with other complications arising from twist. The shape of one facet of the cell might then be constructed from a grid of non -contiguous small circular domains. The geometry of deformation modes within initially circular domains is described in SI Appendix C. Cruciate deformations and neutral axes In several modes of local deformation such as elastic microfibril stretching, extension E(X) in the direction X of the microfibril axis is accompanied by contraction P .E(X) in the direction perpendicular to the microfibrils (Thomas et al. 2021) . The ratio of lateral contraction to axial expansion is the Poisson ratio P , as defined with respect to the microfibril axis (not, here, the cell axis, with respect to which the macroscopic Poisson ratio is defined). Such deformation modes are here termed cruciate, and they are all related in geometry. For cruciate deformation modes it is necessary to define the two orthogonal neutral axes on which elongation and lateral contraction, respectively, are equal to zero. Incorrect designation of these neutral axes mixes fibril shear with the other two deformation modes. The contributions of elongation in the direction of the fibril orientation and lateral contraction are each calculated from the distance from the appropriate neutral ax is. For elongation in the direction of the fibril orientation, the neutral axis is the line normal to the fibrils, passing through the origin at the centre of the rectangle. For lateral contraction the neutral axis is the line parallel to the fibrils, passing through the same origin. Twist differs between the sides and ends of the rectangular domain. Therefore, the rectangle deforms to a parallelogram. The average elongation, narrowing and twist of the rectangle can be deduced from the (x,y) displacements of the centre -points of the initial rectangle’s four sides. Perpendiculars drawn from the neutral axes to these four mid-points form the lines along which axial elongation and lateral contraction of the rectangular domain are measured , except when they extend outside the boundary of the initial rectangle, in which case only that part of the line lying within the rectangle is measured. List of symbols Fractional elongation of cell wall: E Fractional change in width of cell wall: W Axial twist angle (clockwise): t(Ax) Lateral twist angle (clockwise): t(L) .CC-BY-NC-ND 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted December 25, 2025. ; https://doi.org/10.64898/2025.12.23.696159doi: bioRxiv preprint 5 Microfibril angle (angle between cell axis and microfibril axis): θ Fractional elongation along fibril axis (fibril stretching, interdigitated shear , wave straightening): E(X) Fractional expansion at right angles to the fibril axis (widened fibril spacing , wave formation): E(Y) Shear angle (regular shear): s Distance along fibril axis: X

Deformation of straight fibrils Four modes of deformation of an array of straight fibril s are illustrated in Fig.1 and their geometry is examined below. Fig.1. Modes of deformation of a rectangular fibril array in which the fibrils remain straight. A. A cruciate deformation mode (e.g. microfibril stretching) where elongation along the fibril axis is accompanied by orthogonal contraction as described by Poisson ratio P: here P = 0.4. B. Fibril rotation by a (clockwise) angle R . C. Regular shear between fibrils , where each fibril slides in the same direction over the fibril below, defining a shear angle s . D. Widened fibril spacing, by a fraction W. When shear between fibrils was evaluated by (Keckes et al. 2003) , it was assumed to follow the normal engineering description: that is, successive microfibrils are displaced in the same direction by a fixed distance relative to their spacing, and thus a constant shear angle s can be defined. Here, that sliding process is called regular shear . However, the sliding of successive microfibrils need not be identical and thus need not conform to a constant shear angle. During the elongation of wood cell-walls with very low microfibril angle , under the rather high stresses where elastic stretching of cellulose is observed, cellulose stretching (from X-ray diffraction, XRD) did not account for all the macroscopic extension (Thomas et al. 2021). Rotation was negligible and the existence of another kind of fibril sliding was therefore inferred (Thomas et al. 2021). XRD data on primary cell walls at very large elongations allow the same inference (Yu et al. 2024) . In the simplest representation of this second mechanism, the direction of sliding alternates between successive fibrils. Therefore, this deformation mode is here called interdigitated sliding . Interdigitated sliding is an example of a cruciate deformation mode as described above. Regular shear is not. It can be assumed that regular shear and interdigitated s liding are simply the extremes of a continuum of sliding deformations, and that the direction of sliding might be intermediate or random rather than uniform as in regular shear or strictly alternating as in interdigitated sliding. Separating these two forms of sliding is maybe therefore artificial, but it simplifies both the understanding a nd the .CC-BY-NC-ND 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted December 25, 2025. ; https://doi.org/10.64898/2025.12.23.696159doi: bioRxiv preprint 6 geometric analysis of the phenomenon. Both forms of sliding may occur between microfibrils within junction zones of primary walls or between aggregates of microfibrils, as presumed in wood (Thomas et al. 2021) . An alternative way to express the existence of a continuum of sliding modes would be to define a parameter r [0.5 < r < 1 ] representing the proportion of s liding that is unidirectional towards the end of the fibrils nearest the apical end of the cell (right over left, RL in Fig. 1: the opposite direction is left over right, LR) . Then r = RL/(LR+RL) = 1 signifies regular shear and r = 0.5 signifies interdigitated s liding, or random s liding averaging to the same magnitude as interdigitated sliding. It is normal engineering practice to assume that what is here called regular shear is not accompanied by any lateral contraction (Keckes et al. 2003), a convention followed in this study, consistent with constant volume as in solid materials . Note, however, that cell-wall volume is not necessarily constant: water can be removed or supplied from within the cell if the water activity (determined partly by pectins (Jarvis 1992) ) changes. The density of confined water may also change (O'Neill et al. 2017). It can reasonably be suggested that for interdigitated sliding, lateral (Poisson) contraction accompanies elongation along the fibril axis, but the Poisson ratio is difficult to estimate. At very small (probably elastic) interdigitated sliding deformations, the data of (Zhang et al. 2021a) suggest that microfibril spacings should increase slightly, i.e. the Poisson ratio should be negative. For larger deformations where a stick -slip mechanism comes into play (Zhang et al. 2021a), gaps closing behind withdrawn fibril ends might lead to a positive Poisson ratio. Geometry of cruciate deformations in rectangular domains of moderate aspect ratio The deformation modes classed here as cruciate are microfibril stretching, interdigitated sliding and the straightening of waves. For rectangular domains of moderate aspect ratio a (all domains with 0.5 < a < 2, and domains outside this range at high or low microfibril angles), the elongation of the rectangle is measured in the y direction between the midpoints of the top and bottom, and contains contributions from both E(X) in the direction of the fibrils and P .E(X) in the orthogonal direction, each multiplied by the distance of the top and bottom mi dpoints from the corresponding neutral axis (Fig. S1 A). The axial twist is measured from the x coordinate of the same displacement. The change in the width of the rectangle and the lateral twist are calculated in an analogous way from the (x,y) displacement of the mid -points of the sides of the rectangle. (Fig. 2; SI Appendix A and Fig. S1A). Elongation E = E(X)cos2θ – P .E(X)sin2θ Change in width = E(X)sin2θ – P .E(X)cos2θ Axial twist = atan[(E(X)+P .E(X))sinθ.cosθ] Lateral twist = - atan[(E(X)+P .E(X))sinθ.cosθ] .CC-BY-NC-ND 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted December 25, 2025. ; https://doi.org/10.64898/2025.12.23.696159doi: bioRxiv preprint 7 Fig.2. Relationship of changes in array dimensions to microfibril angle for cruciate deformations with Poisson ratio 0.4 and 3.0. A Poisson ratio of 0.4 is potentially representative of microfibril stretching or interdigitated shear. A Poisson ratio of 3.0 is representative of the straightening of waves. Twist has the same magnitude in both axial and lateral directions but is opposite in sign. In consequence the rectangle distorts to a parallelogram (Fig. 1) . That is why the dimensional changes are calculated between the midpoints of the edges. Rotation cannot compensate simul - taneously for the axial and the lateral twist associated with the cruciate deformation modes. Geometry of cruciate deformations in rectangular domains of high and low aspect ratio Different functions apply to very long or very short rectangles, with aspect ratio a outside the range [1/  < a < ] where  = ( tanθ + 1/tanθ) (Fig. S2). Examples of rectangular cell -wall domains with high aspect ratio include the edge regions of elongated primary cell walls (Lee et al. 2023); collenchyma cell walls (Thomas et al. 2013) ; and the tangential walls of softwood tracheids under axial tension (Thomas et al. 2021) . If the microfibril angle is close to 0º or 90º, however, these domains may still be within the range [1/ < a , the point on the neutral axis from which the perpendicular is drawn to the midpoint of each end falls outside the boundary of the rectangle, and the distance from which elongation is measured is calculated from the rectangle’s side border instead: Elongation = E(X)/(a. tan θ) – P .E(X).(tanθ)/a Axial twist = atan[(P+1)E(X)/a] The change in width and the transverse twist are the same as for a rectangle of moderate aspect ratio. Note that in contrast to rectangles of moderate aspect ratio the fractional elongation and axial twist are dependent on a, the fractional elongation decreasing as the aspect ratio increases. .CC-BY-NC-ND 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted December 25, 2025. ; https://doi.org/10.64898/2025.12.23.696159doi: bioRxiv preprint 8 Fig. 3. Discontinuous variation of elongation and change in width on cruciate deformation (P=0.4) of (A) rectangular domains with varying aspect ratio, at microfibril angles of 30º and 70º and (B) Variation with microfibril angle at aspect ratio a = 2.0 (moderate, dotted) or 2.9, which is sufficient to enter the high aspect ratio regime for elongation at microfibril angles much greater or less than 45º. At high aspect ratios there is no effect on width. Data on twist are in Fig. S3 The radial walls of softwood tracheids and other cell walls growing from the vascular cambium are of very low aspect ratio with respect to the axis of growth, although being heterogeneous in microfibril angle they do not strictly conform to the requirements of this study (Sedighi-Gilani, Sunderland, Navi 2005). For a short, wide rectangle with a < 1/, the point on the neutral axis from which the perpendicular is drawn to the midpoint of each side falls outside the rectangle, and the distance from which change in width is measured is calculated from the rectangle’s top or bottom boundary. Change in width = E(X).a.tanθ - P .E(X).a/tanθ Transverse twist = -(E(X) + P .E(X)).a The elongation and the axial twist are the same as for a rectangle of moderate aspect ratio. Rectangles with 0.5 < a < 2 have ‘moderate’ aspect ratio at all values of θ. Outside these limits, the regime is dependent on both θ and a (Figure S 2). That is the origin of the discontinuous behaviour of the elongation and other functions for rectangles of high or low aspect ratio. Regular shear between fibrils Regular shear as defined above has been widely assumed to occur in all cell walls under tension or during growth, although direct experimental demonstration and quantification are quite difficult. Regular shear was used by Keckes et al. (2003) to explain the behaviour of wood having relatively high microfibril angles. The following functions are derived in SI Appendix A (Fig. S1C). Elongation E = tan(s) sinθ.cosθ Change in width = - tan(s) sinθ.cosθ Axial twist = atan[tan(s).sin2θ] Lateral twist = atan[tan(s).cos2θ] The relationship to microfibril angle is shown in Fig. 2. Maximum elongation and minimum width are reached when the microfibril angle ϑ = 45°, as expected for engineering shear. Fibril rotation Fibril reorientation towards the direction of elongation has been observed by XRD (Thomas et al. 2021), vibrational spectroscopy (Guo and Altaner 2019) and atomic force microscopy (AFM) (Yu et al. 2024), both in wood and in primary cell walls, where it accompanies both growth and extension under externally applied .CC-BY-NC-ND 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted December 25, 2025. ; https://doi.org/10.64898/2025.12.23.696159doi: bioRxiv preprint 9 stress (Cosgrove 2024b). Where rotation counteracts twist from other deformation modes it gives an increase in both length and width. If the rotation angle = R Elongation = change in width = 1/cosR - 1 Twist = R in both axial and lateral directions These functions are independent of the aspect ratio of the rectangular domain. In principle rotation can be in either direction. However in practice, rotation towards the direction of stress or growth seems normally to be observed: for example, the increasingly axial microfibril orientation towards the outermost and oldest layers of an elongated primary cell wall (Vian, Roland, Reis 1993). Microfibril stretching Elastic elongation of microfibrils has been observed by XRD and vibrational spectroscopy in wood with low microfibril angle (<10°) (Thomas et al. 2021; Peura et al. 2007) and in axially oriented microfibrils of primary cell walls under external tension (Yu, Zhang, Cosgrove 2024) . Microfibril stretching conforms to the cruciate model of deformation as described above (Fig. 2) . Crystallographic Poisson ratios of approximately 0.5 have been observed for wood cellulose (Altaner et al. 2014) and for intact wood (Ando et al. 2018) , although curiously there seems to be no change in centre-to-centre spacing as observed by SANS (Thomas et al. 2020) . It is assumed that no ways of stretching microfibrils exist additional to those expressed in the longitudinal expansion of the crystallographic unit cell (Thomas et al. 2021). Widened fibril spacing Increased spacing of microfibrils in primary cell-walls has not often been observed but was suggested by Marga et al. (2005). While respacing of wood microfibrils within macrofibrils is readily observed by small- angle neutron scattering ( SANS) (Thomas et al. 2020), increases in macrofibril spacing in wood would be difficult to detect except by nanoscale imaging. Widened fibril spacing (Fig. 2) can be described by a modified version of the cruciate deformation functions with E(X) = 0. If the increase in fibril spacing is denoted as W (instead of P .E(X)) then (SI Appendix A): Elongation = Wsin2ϑ Change in width = Wcos2ϑ Axial twist = -atan (Wsinϑ.cosϑ) Transverse twist = atan (Wsinϑ.cosϑ) Curved fibrils: sine-wave patterns Wave formation in laterally oriented microfibrils has been observed for onion epidermal walls under severe external tension (Yu et al. 2024) . Coarse-grained modelling studies of primary cell -walls (Zhang et al. 2021b) predict that waves will form passively when transversely oriented microfibrils are compressed along the microfibril axis by narrowing of other cell - wall layers . Alternatively, forces applied perpendicular to the microfibril axis at isolated points along each microfibril could also lead to the formation of waves, with maxima and minima at the points of application of force. These mechanisms could occur together when a rectangular domain with high microfibril angle is stretched, and would not be easy to distinguish. Nor could either of these passive mechanisms for wave formation be easily distinguished from any waves that .CC-BY-NC-ND 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted December 25, 2025. ; https://doi.org/10.64898/2025.12.23.696159doi: bioRxiv preprint 10 might result directly from sinuous paths of CSCs across the cell membrane. Similar sinuosity, with lower amplitudes, contributes to the spread of fibril orientations in wood cell -walls (Thomas et al. 2021). Concerted waved patterns of large groups of microfibrils have often been observed by EM at the inner faces of primary cell-walls (Marga et al. 2005; MacKinnon et al. 2006; Yu et al. 2024) , although there is some doubt as to their origin. They could be a consequence of the relaxation of turgor stress, or due to shrinkage on drying prior to imaging. When waves in axially oriented fibrils are initially present in a cell -wall under external tension, they tend to be pulled straight (Yu et al. 2024), as predicted from coarse -grained modelling (Zhang, Yu, Cosgrove 2025; Zhang et al. 2021b). It is not clear if that such waves are available to be straightened during growth. A single waved fibril can be approximated as a sinewave with amplitude 2A in Cartesian coordinates where X = distance along the axis of the wave, and the axis is oriented at microfibril angle = θ to the cell axis. It is assumed that when the microfibril straightens, θ remains constant, i.e. the local strain is along the microfibril axis not the cell axis. There is support for this view from in experiments on wood (Jarvis 2024) and in coarse-grained ( CG) modelling of primary walls (Zhang, Yu, Cosgrove 2025) . Wave formation is simply the reverse of straightening. Geometry of wave formation and straightening Straightening a wave leads to elongation E(X) along the microfibril axis . The geometry of wave formation being the reverse of the geometry of wave straightening, E(X) for wave formation is negative. The dependence of E(X) on the wave amplitude is derived in SI Appendix B and is non-linear, approximated as E(X) = 0.1982A2 + 0.0201A. The associated lateral contraction on wave straightening depends on how the fibrils are arranged or bundled. If separate microfibrils are sinuous and randomly arranged, with the same overall orientation but with the waves varying in amplitude and out of phase (Fig. S5A), the lateral contraction on straightening could be as much as the mean amplitude of the waves. That assumes, however, that the fibrils remain at the same spacing when they straighten. They could alternatively be held apart by non -cellulosic polymers. Consistent with th e second alternative, straightened microfibrils of epidermal cell - walls under extreme tensile strain settled to a relatively uniform 7 nm spacing (Yu et al. 2024), too far apart for lateral association. In contrast, if adjacent microfibrils are waved in-phase and possibly, although not necessarily, bundled, the lateral contraction on straightening may be small. In intermediate cases the lateral contraction is a function of the amplitude A and the coherence length C of the wave pattern normal to the microfibril axis . For individually sinuous microfibrils C is negligibly small. For concerted wave patterns C is larger, e.g. visually of the same order of magnitude as the wavelength in micrographs such as those of (MacKinnon et al. 2006). The array then contracts laterally on straightening from approximately (C+A) to C, and the fractional change in width P.E(X) = A/(C+A). For low-amplitude waves where E(X) is very small compared to the amplitude A , for the straightening of non-coherent waves the apparent Poisson ratio P can be much greater than unity (Fig. S7). The geometry of wave formation, or of an increase in wave amplitude, is then a close approximation to .CC-BY-NC-ND 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted December 25, 2025. ; https://doi.org/10.64898/2025.12.23.696159doi: bioRxiv preprint 11 the geometry of widened fibril spacing but with the axes transposed. When P>>1, if waves are created by local forces orthogonal to the fibrils at the maxima and minima of the resulting waves, these forces will exert a large leverage d amplification (mechanical advantage) = P on the force of contraction along the fibril axis. The straightening of waves, concerted or not, is an example of a cruciate deformation and can be described by the functions mentioned above, with Poisson ratios higher than in other cruciate deformation modes. At the nanoscale, wave formation and straightening entail smaller rearrangements of local structure than sliding deformations, and have been simulated to occur before sliding deformations (Zhang et al. 2021b) . There is a small amount of sliding or respacing due to the curvature of concerted waves (SI Appendix 2). The anastomosing microfibril network of primary cell-walls may be approximated as a sinuous, non -coherent, out -of-phase array in which occasional junction zones arise when successive microfibrils touch (Cosgrove 2024b) (Fig. S5A). When such an array is straightened, microfibrils forming large-amplitude waves contract much more than initially straighter microfibrils. The consequence is either a dispersion of sliding deformations among junction zones, with the junction zones connecting to large - amplitude waves deforming the most : or if local sliding at the junction zones does not occur, a dispersion of tensile strain in the fibril segments between them. Macrofibril networks in secondary cell walls are topologically similar, and in woo d with low microfibril angle under tension, the tensile strain demonstrated by the cellulose diffraction patterns became more disperse as strain increased (Thomas et al. 2021).

This analysis was strictly limited in its objectives, particularly through the absence of any quantitative attribution of forces , moduli or reversibility of local deformation, and through the restriction to a single lamella of a primary cell -wall or to the S2 lamella of a wood cell -wall. The outcome is not, therefore, a ‘model’ of tensile elongation or growth. In both growing plant tissues and wood, the behaviour of the whole cell -wall cannot be fully predicted from the sum of the behaviours of its la mellae because each lamella constrains the deformation of the others (Guo, Altaner, Jarvis 2020; Zhang et al. 2021b). Different types of fibril sliding and their control A key finding was the need to propose two kinds of sliding deformation between fibrils, differentiating between what are here called regular shear and interdigitated sliding - although these sliding motions are recognised to represent the two extremes of a continuum. Regular shear and interdigitated sliding can both contribute to the elongation of a rectangular domain, but have different outcomes for width and twist. Under uniaxial tension the projected force is greater for RL sliding, as in regular shear, to an extent that depends on the microfibril angle and is maximal at ϑ = 45º. That is not the case u nder biaxial turgor-driven tension in growing primary cell-walls. In primary cell -walls with a network of sinuous microfibrils associating at localised junction zones, the local RL/(LR+RL) ratio .CC-BY-NC-ND 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted December 25, 2025. ; https://doi.org/10.64898/2025.12.23.696159doi: bioRxiv preprint 12 depends on how the network geometry puts tension on each fibril. T he balance between RL and LR sliding may also be influenced by the structure of the junction zones and by any associated enzymes. Interfaces between ‘hydrophobic’ 200 faces of the native cellulose structure show easier and smoother simulated stick -slip sli ding than interfaces between hydrogen -bonded ‘hydrophilic’ faces (Zhang et al. 2021a). In these simulations there was little difference between parallel and antiparallel pairs of fibrils. However, it may be envisaged that the binding of expansins (Cosgrove 2024a) would differ between parallel and antiparallel interfaces. Thus, the direction of travel of cellulose synthase complexes laying down adjacent microfibrils might determine the affinity of expansins for their interfaces, contributing to the formation of local ‘hotspots’ (Cosgrove 2024b) at which sliding is favoured - in one direction more than the other, from the chirality of expansin binding – and thus influencing the ratio of RL and LR sliding modes . Mechanisms like these might underlie how a growing cell changes shape as it grows. In wood under tensile stress, enzymatically facilitated sliding is absent but otherwise, similar considerations may determine the dependence of the tensile moduli and macroscopic Poisson ratio on microfibril angle (Thomas et al. 2021) . Sliding is presumably between macrofibrils or larger units, not microfibrils. Curled, tapered strands observed at tensile fracture surfaces (Guo, Altaner,Jarvis 2020) may indicate locked-in shear between units larger than macrofibrils. Waves and changes in fibril spacing The principal direction of local growth is commonly observed to be at right angles to the mean fibril orientation in primary cell - walls (Cosgrove 2024b) . That would seem natural if the predominant deformation mode were an increase in fibril spacing, but imaging and coarse -grained modelling studies seem to imply that sliding motions are more important than respacing (Cosgrove 2024b). The formation or straightening of wave patterns, especially if their coherence length is short, entails local respacing of fibrils whether or not the mean spacing changes . Formation or increased amplitude of non - coherent waves that are initially of low amplitude, with little contraction along the fibril axis, might be difficult to distinguish from widened fibril spacing. Distributed spacings are not readily measurable by SANS (Kennedy et al. 2007). It is difficult to comment on the ease or difficulty of fibril respacing without being able to specify the interaction potential between fibrils. It has been suggested that a Lennard-Jones type of potential function describes microfibril spacings in primary cell-walls, with attractive and repulsive terms and a term describing contact (Kennedy et al. 2007). The structure of water inserting itself between two cellulose surfaces should contribute to such a function (O'Neill et al. 2017; Zhang et al. 2021a), and its contribution should be modulated by the nature of the surfaces and by any pectic or xyloglucan chains that also find room for insertion (Cosgrove 2024b; Jarvis 1992; Zhang, Yu, Cosgrove 2025) . However, too much remains to be experimentally demonstrated about these phenomena, even the location of the relevant polymers. When the mean spacing in a primary-wall microfibril network expands with ingress of water, that could mean either that the wider .CC-BY-NC-ND 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted December 25, 2025. ; https://doi.org/10.64898/2025.12.23.696159doi: bioRxiv preprint 13 spacings would enlarge or that some of the junction zones would peel apart. The observation that primary cell-walls are not weakened as they elongate (Cosgrove 2024b) implies that reduction in the net number of junction zones is not usually a major contributor. The junction zones will then be subjected to varying magnitudes of shear stress, depending non -linearly on the amplitude of the waves that each junction zone connects (Fig. S 5). In extreme situations, junction zones may peel apart. Axially oriented microfibrils are observed to become separated at a more uniform ~7 nm spacing as they are stretched (Yu et al. 2024), presumably with separation of junction zones. I nterdigitated shear would then be facilitated. That could explain why axial microfibril orientation in older layers of primary cell walls does not have the anticipated effect of stopping growth. The influence of aspect ratio An unforeseen outcome was the discontinuous dependence of elongation, width and twist on the aspect ratio of the rectangle. It is noticeable that many of the cell facets in simple, elongating plant tissues fall within the ‘moderate’ range of aspect ratios, where elongation is maximal relative to the required magnitude of the local sliding deformation s. The longitudinal edge domains of these cells, despite high aspect ratio, may still fall within the ‘moderate’ range due to microfibril angles near 90º (Lee et al. 2023) . Where these microfibrils run continuously across the edge region, maintaining their microfibril orientation may require coordinated twist in the two adjacent cell faces. Other very long cells, such as cotton hairs, show tip growth (Seagull 1993) or are discontinuous with respect to microfibril angle (Sedighi-Gilani, Sunderland, Navi 2005). Combining deformation modes The geometric descriptions of individual, nanoscale deformation modes are a prerequisite for calculating how multiple modes interact to allow cell elongation within specific constraints on the width and twist of the cell wall as a whole. The extent of reversibility of each deformation mode (Chen et al., 2025) influences how reversible the whole expansion process will be . These interactions are key to understanding how the shape of growing cells emerges from the interplay of local deformations.

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It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted December 25, 2025. ; https://doi.org/10.64898/2025.12.23.696159doi: bioRxiv preprint 17 .CC-BY-NC-ND 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted December 25, 2025. ; https://doi.org/10.64898/2025.12.23.696159doi: bioRxiv preprint