bioRxiv preprint doi: https://doi.org/10.1101/2020.07.30.228973; this version posted July 31, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-ND 4.0 International license. Multiscale integration of environmental stimuli in plant tropism produces complex behaviors Derek E. Moulton1, Hadrien Oliveri, and Alain Goriely1 Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom This manuscript was compiled on July 30, 2020 1 Plant tropism refers to the directed movement of an organ or or- 2 ganism in response to external stimuli. Typically, these stimuli in- (A) (B) 3 duce hormone transport that triggers cell growth or deformation. In 4 turn, these local cellular changes create mechanical forces on the 5 plant tissue that are balanced by an overall deformation of the organ, 6 hence changing its orientation with respect to the stimuli. This com- 7 plex feedback mechanism takes place in a three-dimensional grow- 8 ing plant with varying stimuli depending on the environment. We 9 model this multiscale process in filamentary organs for an arbitrary (C) 10 stimulus by linking explicitly hormone transport to local tissue de- 11 formation leading to the generation of mechanical forces and the 12 deformation of the organ in three dimensions. We show, as exam- 13 ples, that the gravitropic, phototropic, nutational, and thigmotropic 14 dynamic responses can be easily captured by this framework. Fur- 15 ther, the integration of evolving stimuli and/or multiple contradictory 16 stimuli can lead to complex behavior such as sun following, canopy 17 escape, and plant twining. Plant tropism | Biomechanics | Morphoelasticity | Rod theory | Mathe- Fig. 1. Classic experiments on tropic responses. (A) Gravitropism: a potted plant matical model realigns itself with gravity (13). (B) Thigmotropism: a twining vine develops curvature when in contact with a pole (6). (C) Phototropism: a plant reorients itself towards the light source (18th Century experiments by Bonnet (14), correctly interpreted by 1 lant tropism is the general phenomenon of directed growth Duhamel du Monceau (15, 16)). 2 Pand deformation in response to stimuli. It includes pho- 3 totropism, a reaction to light (1); gravitropism, the reaction 4 to gravity (2, 3); and thigmotropism, a response to contact typically receives multiple stimuli at the same time and in 32 5 (4), among many others (see Fig. 1). The study of tropisms different locations (12). The resulting movement is an integra- 33 6 in plants dates back to the pioneering work of giants such as tion of multiple signals and movements. Third, any tropism 34 7 Darwin (5) and Sachs (6), and has been a central topic for is fundamentally a multiscale phenomenon. Transduction of 35 8 our understanding of plant physiology ever since. Tropisms an environmental cue takes place from the organ to the cell 36 9 form a cornerstone subject of modern plant biomechanics (7), and involves, ultimately, molecular processes. A hormonal 37 10 crop management strategies (8), as well as systems biology response is induced, which leads to different cells expanding 38 11 and plant genomics (9). Being sessile by nature, plants lack at different rates in response to the chemical and molecular 39 12 the option to migrate and must adapt to their ever-changing signals. However, one cannot understand the change in shape 40 13 environment. The growth response of individual plants to of the plant and its position in relation to the direction of the 41 14 environmental cues will determine the yield of a crop in unusu- environmental stimulus at this level. To assess the effectiveness 42 15 ally windy conditions, will decide the future of rainforests in a of the growth response, one needs to zoom out. The net effect 43 16 world driven by climate change, and may be key for colonizing of a non-uniform cell expansion due to hormone signaling is a 44 17 foreign environments such as Mars. tissue-level differential growth (1) as depicted in Fig. 2. At the 45 18 Mathematical modeling plays an invaluable role in gaining tissue level, each cross section of the plant can be viewed as 46 19 a better understanding of tropisms and how plants may re- a continuum of material that undergoes non-uniform growth 47 20 spond to a change in their environment (10). Yet, a general and/or remodeling (17). Differential growth locally creates 48 21 mathematical description of tropisms is a grand challenge. curvature and torsion, but it also generates residual stress 49 22 First, the growth response tends to be dynamically varying: (18). In this way the global shape of the plant, as well as its 50 23 a sunflower grows to face the sun, but as it grows the sun material properties, evolve. To characterize this global change, 51 24 moves, so the environmental influence – the intensity of light and to update the position of the plant in the external field, a 52 25 impacting on each side of the sunflower – is changing during further zooming out to the plant or organ level is appropriate. 53 26 the process. Similarly, a tree branch may align with the ver- At the plant level, the global shape, material properties, and 54 27 tical in a gravitropic response; decreasing the likelihood of positioning in the external stimulus are well described by a 55 28 breaking under self-weight; however, the growth response itself physical filament: here, the plant is viewed as a space curve 56 29 may increase the branch weight and thus change the stimulus endowed with physical properties dictated by the lower level 57 30 (11). Second, while there exist numerous experimental setups tissue scale, and its shape and motion can be described by 58 31 that enable to carefully isolate a particular stimulus, a plant the theory of elastic rods, that has been applied to multiple 59 www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX PNAS | July 30, 2020 | vol. XXX | no. XX | 1–10 bioRxiv preprint doi: https://doi.org/10.1101/2020.07.30.228973; this version posted July 31, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-ND 4.0 International license. 60 biological contexts, from DNA and proteins, to physiology and view of a plant as a problem-solving control system that is 99 61 morphogenesis (19–22). actively responding to its environment. 100 62 The challenge of formulating a mathematical model of 63 tropism is further complicated by the extraordinary variation 64 in plants and the multiple types of tropism. Within a single 65 plant, a tropic response may refer to the growth and movement 66 of the entire plant, or a subset: a single branch, vine, stem, or 67 root. Here, we use the word ‘plant’ to refer to the entire class 68 of plant structures that may undergo such growth responses. 69 Moreover, even within a single plant, multiple environmental 70 cues will combine and overlap in effecting mechanotransductive 71 signals, hormonal response, differential growth, and ultimate 72 change in shape (23); e.g. a sunflower exhibiting phototropism 73 still perceives a gravitational signal. 74 At the theoretical level, a variety of approaches have re- 75 cently been proposed. Growth kinematics models successfully 76 describe the tropic response at the plant-level (7, 24, 25), but 77 do not include mechanics and cellular activities. A number of 78 elastic rod descriptions of tropic plant growth have also been 79 proposed (26–31); these involve a full mechanical description 80 at the plant-level, with phenomenological laws for the dynamic 81 updating of intrinsic properties such as bending stiffness and 82 curvature, but specific cell- and tissue-level mechanisms are not 83 included. Multiscale formulations have also appeared, includ- 84 ing Functional-Structural Plant Models (8, 32, 33) and hybrid Fig. 2. Tropism is a multiscale dynamic process: the stimulus takes place at the plant 85 models with vertex-based cell descriptions (34, 35). These or organ level and its information is transduced down to the cellular level creating a tissue response through shape inducing mechanical forces that change the shape 86 computational approaches have the potential to incorporate of the organ. In the process, the plant reorients itself and, accordingly, the stimulus 87 effects across scales but are limited to small deformations changes dynamically. 88 compared to the ones observed in nature. 89 The goal of this paper is to provide a robust mathematical 90 theory that links scales and can easily be adapted to simulate 1. Multiscale modeling framework 101 91 and analyze a large number of overlapping tropisms for a 92 spectrum of plant types. Our mathematical and computational The key to our multiscale approach is to join three different 102 93 framework includes (i) large deformations with changes of scales: stimulus-driven auxin transport at the cellular level; 103 94 curvature and torsion in three-dimensional space; (ii) internal tissue-level growth mechanics; and organ-level rod mechanics. 104 95 and external mechanical effects such as internal stresses, self- 96 weight, and contact; and (iii) tissue-level transport of growth A. Geometric description of the plant. We start at the organ 97 hormone driven by environmental signals. By considering the scale and model the plant as a growing, inextensible, unshear- 98 integration of multiple conflicting signals, we also provide a able elastic rod. This is a one-dimensional filamentary object that can bend and twist with some penalty energy, but also ex- 3 tend without stretching by addition of mass.