Multiscale Integration of Environmental Stimuli in Plant Tropism Produces Complex Behaviors

Home , Thigmotropism, Tropism

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 The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.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

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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. Let r(S, t) ∈ R describe its centerline, where S is the initial arc length mea- Significance Statement sured from the base of the plant towards its tip (see Fig. 3(A)). To survive and to thrive, plants rely on their ability to sense Along this centerline, we define an orthonormal basis {di}, multiple environmental signals, such as gravity or light, and i = 1, 2, 3, oriented such that d3 aligns with the tangent ∂r/∂S respond to them by growing and changing their shape. To do in the direction of increasing S, and (d1, d2) denote material so, the signals must be transduced down to the cellular level directions fixed within each cross-section. From the director to create the physical deformations leading to shape changes. basis, the Darboux vector is defined as u = u1d1 +u2d2 +u3d3, We propose a multiscale theory of tropism that takes multiple and encodes the rod’s curvature, torsion and twist (17). For a stimuli and transforms them into auxin transport that drives given curvature vector, the shape of the rod is determined by tissue-level growth and remodeling, thus modifying the plant integrating the system of equations shape and position with respect to the stimuli. This feedback loop can be dynamically updated to understand the response to ∂r ∂di = γd3, = γu × di, i = 1, 2, 3. [1] individual stimuli or the complex behavior generated by multiple ∂S ∂S stimuli such as canopy escape or pole wrapping for climbing Here γ := ∂s/∂S denotes the total axial growth stretch of each 105 plants. section mapping the initial arc length S to the current arc 106 length s (36). At each value of S, the cross section is defined by 107 All authors have contributed equally. 2 a region (x1, x2) ∈ ΩS ⊂ R , where x1, x2 are local variables 108 The authors declare no conflict of interest. describing the location of material points in the respective 109

1To whom correspondence should be addressed. E-mail: [email protected], directions d1, d2. In terms of the local geometry, any material 110 [email protected] point X = (X1,X2,X3) in the plant can be represented by its 111

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Physically, stimuli are fields acting in space at a point 127 3 X = X1e1 + X2e2 + X3e3 ∈ R and changing over time t. 128 They can be either scalar fields, f = f(X, t), e.g. chemical, 129 temperature, or light intensity; vector fields, F = F(X, t), e.g. 130 geomagnetic field, gravity, or light direction; and, possibly, 131 tensor fields (e.g. mechanical stress–not considered here). 132 These stimuli are in general functions of both space and time 133 which makes plant tropism a physical theory of fields (which 134 is appropriate since plants grow in physical fields). Since 135 the stimulus is defined at points in space, we must take into 136 account the orientation and the position of the plant in space. 137 For example, the cellular response to light in phototropism is 138 linked to the relative orientation of the plant in relation to 139 the light source. In the case of a vector stimulus F, we must 140 therefore decompose the stimulus in the local basis: 141

F = F1d1 + F2d2 + F3d3,Fi = F · di, i = 1, 2, 3. [3] 142

Next, we link an external stimulus to the cellular response. 143

C. Cellular response and auxin transport. At the cellular level, 144 deformation takes place through an anelastic expansion of the 145 cell walls in response to turgor-induced tension (38, 39). We 146 refer to any geometric change of cellular shape as growth. 147 While detailed models of these cellular processes are available 148 (40–42), we adopt here a coarse-grained view in which the 149 anelastic expansions are connected locally via a single hor- 150 mone concentration ﬁeld that plays the role of a morphogen 151 (for simplicity, we use the phytohormone auxin, but any other 152 hormone could be used, or, more realistically, a combination 153 of hormones, interacting with heterogeneous gene expression 154 patterns). Indeed, laterally asymmetrical auxin redistribu- 155 tion is broadly accepted as a universal mechanism underlying 156 tropisms (43, 44). A lateral gradient is controlled via the relo- 157 calization of auxin transporters in response to tropic signals 158

Fig. 3. (A) Each cross section Ωs of the rod is parametrized with its arc length s (45). In shoots, higher levels of auxin are generally associated 159 (oriented acropetally) and equipped with a local material basis {d1, d2, d3}. (B) with faster growth. The resulting asymmetrical growth of cells 160 Gravitropism: the gravity vector (1) is sensed in each cross section and causes lateral elicits global curvature at the organism level through pathways 161 auxin flow (2). (C) Phototropism: the light vector is sensed at the plant apex and that are not completely understood (46). Therefore, in our 162 results in the establishment of an apical auxin profile (1) that is transported basipetally with attenuation (2). (D) The circumnutation is generated by an internal oscillator with model auxin flux is a function of tropic signal and growth is 163 pulsation ω associated with rotating auxin profile at the apex (1). The apical profile is taken to depend only on auxin concentration. 164 transported basipetally (2), generating curvature and torsion. (E) Thigmotropic pole We model the effective migration, synthesis and uptake of 165 wrapping is triggered by a contact (1) eliciting an asymmetrical auxin profile (2), which auxin through a standard reaction-advection-diffusion equation 166 is in turn transported helically (3) to the rest of the plant with signal attenuation (4). (47) for the auxin concentration A(x1, x2, S, t): 167

∂A stim + ∇ · J = −QA + C, J = −κ∇A + J [4] 168 112 arc length S and its position (x1, x2) on the cross section at ∂t 113 S as follows: where J is the auxin ﬂux, Q is the uptake of auxin, and C 169

114 X = r(S, t) + x1d1(S, t) + x2d2(S, t), for (x1, x2) ∈ ΩS . [2] captures any sources or sinks. The flux is a sum of a diffusive 170 diff stim component J = −κ∇A, and a stimulus component J . 171 115 We can now use this representation to formulate the stimuli. Depending on the particular tropism, the information about 172 stim the stimuli is contained either in J , or in a boundary or 173 116 B. The stimuli. Tropic stimuli are characterized by their origin, source term. The auxin transport equation Eq. (4) is combined 174 117 sign, and direction (37). Signal origin includes: chemicals, with a no-flux condition J · n = 0 at the outer boundary of 175 118 water, humidity, gravity, temperature, magnetic fields, light, each cross-section, where n is an outward normal vector to 176 119 touch. Tropisms can have a sign: positive if the plant grows the boundary ∂ΩS . 177 120 towards or in the direction of the stimulus or negative if it 121 moves away from the stimulus. The direction of tropism D. Tissue-level growth and remodeling. Once the auxin dis- 178 122 describes the orientation of the response with respect to a tribution is known from the solution of Eq. (4), we can relate 179 123 directed stimuli: exotropism is the continuation of motion in the growth field at the tissue level to the concentration A. 180 124 the previously established direction, orthotropism is the motion Here, we use the general theory of morphoelasticity (17) that 181 125 in the same line of action as the stimulus, and plagiotropism, assigns at each point of the plant a growth tensor dictating 182 126 is the motion at an angle to a line of stimulus. the deformation due to growth. Physically, this tensor field 183

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184 integrates the multiple contributions of local pressure, cell F. Rod mechanics sets the plant position in the stimulus field. 215 185 material properties and tissue geometry, all regulated via the Once the intrinsic curvatures and elongation of the plant 216 186 cell metabolic and genetic activity into a single object de- following growth have been updated, the plant position and 217 187 scribing the local change of shape of an elementary volume orientation are updated by solving the Kirchhoff equations 218 188 element (48). This growth tensor may be different in different (36, 50) for the balance of linear and angular momentum for 219 189 directions (anisotropic growth) and/or spatially varying (het- given external forces such as self-weight or wind, and subject 220 190 erogeneous growth) (49). Since we are interested in change of to any kinematic or contact constraints (see SI Section 3 for 221 191 curvature and torsion, we assume that growth only takes place, details). 222 192 locally, along the axial direction and not in the cross-sectional Since at this scale the plant is treated as a one-dimensional 223 193 direction (hence prohibiting a change in thickness). In that structure, its equilibrium shape is easily computed even in 224 194 case, the only non-trivial component of the growth tensor is complex non-planar geometries. Once the deformation is 225 195 a single function g(x1, x2, S, t) that describes the change of determined, the multiscale cycle is completed by updating the 226 196 axial length of an infinitesimal volume element (see SI Section map between external stimulus and cell-scale response with 227 197 1). An initially straight filament of length L0 with g constant respect to the updated orientation, and the process repeats. 228 198 at all points would grow to a new straight filament of length 199 L = L0(1 + g) (18). If, however, g varies from point to point, G. Summary. The multiscale flow of information for a given 229 200 the same filament would tend to bend and twist as shown in stimulus field proceeds as follows: 230 201 Fig.2. (I) given an initial plant shape, a stimulus F impacts auxin 231 202 Next, we connect the axial growth function g to the con- transport and thus local concentrations of auxin via the 232 203 centration of auxin A via a growth law of the form transport equation (4); 233

∂g ∗ (II) the local auxin concentration A changes the local growth 234 204 = β(A − A ) − ξg, [5] ∂t ﬁeld that impacts the intrinsic curvatures and axial ex- 235 tension of the plant via Eqs. (7)–(10); 236 ∗ 205 where A is a baseline level of auxin, and β characterizes the (III) the new intrinsic curvatures and external conditions de- 237 206 rate at which an increase in auxin generates growth. The extra termine the new mechanical equilibrium of the plant, thus 238 207 term −ξg models autotropism, the tendency to grow straight. changing the plant position and shape in the stimulus 239 208 Indeed, this term introduces a decay of the signal in time so ﬁeld. 240 209 that the total growth stretch is a function of a time-integrated

210 auxin signal To illustrate how complex plant morphologies and tropic 241 responses may be eﬃciently simulated and analyzed within 242 t Z this framework, we show how this multiscale process can be 243 ˜ ∗ −ξ(t−t˜) ˜ 211 g(x1, x2, S, t)= β (A(x1, x2, S, t) − A )e dt. [6] applied to a diversity of tropic responses and combined stimuli. 244 −∞

2. Examples 245 E. Change in local shape and properties. The axial growth function g is defined at the tissue scale and, as such, does not A. Gravitropism. Gravitropism has been extensively studied 246 directly give the change in curvature and torsion of the plant. both experimentally and theoretically. The classic description 247 Indeed, the change of shape depends not only on g but also is called the ‘sine rule’ in which the change in curvature follows 248 on the internal mechanical balance of the forces generated by the sine of the angle with the direction of gravity (51). While 249 each growing volume element. Following the general theory it is successful in capturing observed behavior in gravitropic 250 given in (18) and its adaptation to the particular case of plants experiments, it is mostly phenomenological and is only appli- 251 given by the growth law Eq. (5), we compute the intrinsic cable to planar geometry. Here, we show that the sine rule 252 curvatures and elongation of the growing plant (SI Section 2). emerges naturally from our formulation but that it can be 253 In the absence of autotropism (ξ = 0), these curvatures (given generalized to include three-dimensional deformations that are 254 by the vector uˆ) define the shape of the plant in the absence generated when a plant is rotated in time. 255 of body force and external loads: The stimulus for negative gravitropism is the vector field F = −Ge3 which can be written in the plant frame of reference Z as F = f +f3d3 where f := f1d1 +f2d2 is the gravity force acting ∂uˆ1 I = β x2A dx1dx2, [7] in the plane of the cross-section. As it has been proposed that ∂t ΩS plants are insensitive to the strength of gravitational field, it is Z ∂uˆ2 sufficient to use a unit vector representing only the direction of I = −β x A dx dx , [8] ∂t 1 1 2 gravity, i.e. we scale the gravitational acceleration G to 1. If ΩS f ≡ 0, no tropic response will occur. Gravity perception relies ∂uˆ 3 = 0, [9] on specific cells called statocytes distributed along the shoot ∂t (52). Statocytes contain dense organelles (statoliths) that ∂γ Z A = β (A − A∗) dx dx . [10] sediment under the effect of gravity. Tilting of the plant causes ∂t 1 2 ΩS statoliths to avalanche and to form a free surface perpendicular to the gravity vector, providing orientational information to 212 Here and for the rest of the paper, we have assumed that the the cell (53). It has been observed that the gravitropic response 2 213 cross section is circular with radius R, area A = πR /2 and depends upon the angle between the statoliths free surface 4 214 second moment of area I = πR /4. and the vertical, but not on the intensity of the gravitational

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field and the pressure of statoliths against the cell membrane (D) (54). A possible mechanism is that the contact between the (A) statoliths and the cell membrane may trigger relocalization of PIN membrane transporters and a redirection of auxin flux (53). Here, we follow this hypothesis and, accordingly, assume that gravity drives an advective flow of auxin Jstim = kAf. If the statocytes are uniformly distributed within the stem (B) volume then k is constant. We assume also a source and sink 1.0 (E) of auxin at each cross-section, representing a continual axial 0.8 auxin flow, and that auxin transport is advection-dominated 0.6 0.4

with timescales much shorter than the one associated with Alignment growth. Combining the transport equation (4), growth law 0.2 (5) in the absence of autotropism (ξ = 0), and the evolution 0.5 1.0 1.5 2.0 2.5 3.0 t laws given by Eqs. (7)–(10), we obtain (see SI Section 4) the (C) 10 300 (F)

gravitropic curvature law and axial extension law: 250 8 200 6 ∂uˆ1 ∂uˆ2 150 = Cgravf2, = −Cgravf1, [11] 4 100 Torsion

∂t ∂t Curvature 2 50 ∂γ ∆C ∗ 0 0 = β − A . [12] 0.0 0.5 1.0 1.5 2.0 2.5 ∂t QA t

2 Fig. 4. Gravitropism with a rotating base. (A) A base is tilted with respect to the 256 Here, Cgrav = βk∆C/(IQ ) is a single constant characterizing vertical and then rotated about the axis with speed so that one revolution is completed 257 the rate of change of curvature due to gravity and associated every time unit. Gravitropic response is simulated for varying values of gravitropic 258 with a timescale of gravitropic reaction t = 1/ (LC ) , grav grav sensitivity Cgrav = 0.1 (black), 1 (red), 10 (green), and 50 (blue). Alignment with 259 where L is a characteristic axial length, say the length of the the vertical (B) and curvature and torsion (C) are plotted against time for three base 260 plant. Note that the right hand side of Eq. (12) is proportional rotations. Snapshots for cases of (D) slow reaction, Cgrav = 1, (E) intermediate ∗ reaction, Cgrav = 10, and (F) fast reaction, Cgrav = 50. The sequence is read left to 261 to the net ‘excess auxin’, the integral of (A − A ) over the ∗ right, top to bottom, and the base rotation is counterclockwise. Further simulation 262 section, while the quantity A does not appear in Eq. (11). details and parameters provided in SI Section 7. 263 Thus, an excess of auxin is needed for axial growth, while only 264 a redistribution of auxin is needed to generate curvature.

265 In the particular case where the plant can only bend around in Fig. 4(B)-(C). Since the plant’s response time is much slower 295 266 the single axis d2 and all external forces can be neglected, we than the base rotation, the gravitropic response is averaged out 296 267 have f2 ≡ 0, uˆ1 ≡ 0 and the curvature u = uˆ2. Defining α to and the plant hardly deviates from the straight configuration, 297 268 be the inclination angle, Eq. (11) reads never improving its alignment and generating effectively no 298 curvature or torsion. The plant is almost perfectly straight at 299 ∂u 269 = −C sin α, [13] all times (snapshots not included). The red curves denote a 300 ∂t grav case with increased but still small reaction (fast base), which 301

270 which is the classic and widely-used sine law of gravitropism generates only small oscillations in alignment and curvature. 302 271 (24). In this regime, the plant is eﬀectively ‘confused’; the local 303

272 The general laws Eqs. (11) and (12) can be used for more gravitational field is changing too quickly for the plant to make 304 273 complex gravitropic scenarios. Consider, for instance, an any progress towards alignment with the gravitational field. 305 274 experiment in which the base of the plant stem is at a fixed As the reaction rate is increased (or the base rotation 306 275 angle θ from the horizontal and the base is rotated, as shown decreased), interesting morphologies emerge. In the case of 307 276 in Fig. 4(A). Then, in the frame of reference of the plant, the intermediate reaction rate Cgrav = 10 (green curves), the 308 277 the direction of gravity is constantly changing. Here, we plant begins to curve towards the vertical during the first 309 278 consider the case of zero-axial growth and neglect self-weight quarter rotation of the base, bending about the d2 axis and 310 279 (see Section C for these extra effects). The tropic response increasing its alignment. However, as the base continues to 311 280 will generate curvature and torsion depending on the angle rotate, the curvature initially developed has the tip pointing 312 281 and the rotational velocity of the base as shown in Fig. 4. away from the vertical, so the alignment decreases, and the 313 282 For visualisation purposes, we fix the base rotation rate to plant now must bend about the orthogonal d1 axis. As the 314 283 one turn per unit time and vary the tropic reaction rate of base completes its first rotation and the ‘desired’ axis for 315 284 the plant, which is equivalent to a fixed reaction rate and bending returns to the original d2, an inversion occurs (more 316 285 varying base rotation rate via a rescaling of time. In Fig. 4, we visible in the movies provided in SI), creating a large spike in 317 286 simulate three full rotations of the base with varying reaction torsion. This basic process repeats with each rotation. 318

287 rates (see also SI movies 1-4). The evolving morphology is Finally, increasing the reaction rate (or slowing the base) 319 288 characterized by three metrics: an alignment metric in (B) further creates highly complex morphologies as evidenced by 320 289 that measures how closely aligned with the vertical the plant the blue curves. Here the plant quickly aligns with gravity 321 290 is (a value of one is attained if the entire plant is vertical), and and attains near perfect alignment in the first tenth of the 322 291 curvature and torsion in (C) that broadly measure deviation first rotation. As the base rotates away from this aligned 323 292 from a straight configuration (details in SI Section 5). state, we see an interesting phenomenon: the tip of the plant 324 293 Consider first the slowest reaction morphology (equivalent is able to react and maintain alignment with the vertical, but 325 294 to the case of fastest base rotation), given by the black curves since the base of the plant is clamped at an ever-changing 326

Moulton et al. PNAS | July 30, 2020 | vol. XXX | no. XX | 5 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.

327 angle, a loop forms starting at the base and working its way time is defined as tphoto := 1/ (LCphoto). The exponential 375 328 to the tip. This is accompanied by strong variations in the decay in Eq. (16) is due to the uptake of auxin so that less is 376 329 total alignment and increasingly high curvature, with repeated available at the base, while the time shift ttran := L/U of Atip 377 330 spikes in torsion as extra twist is removed. Our simulations accounts for the transport time to the section at arc length 378 331 of this case beyond three rotations suggest that while the s, leading to time-delay differential equations. The change of 379 332 basic process of loops generated at the base and working to curvature thus depends on three quantities: (i) the orientation 380 333 the tip continues, the morphology does not settle down into a of the tip with respect to the light source t = (` − s)/U ago; 381 334 fixed oscillatory pattern, highlighting the potential for complex (ii) the amount of auxin available for the phototropic signal, 382 335 dynamics generated by this highly nonlinear system. which depends on the uptake Q; and (iii) the plant’s response 383 sensitivity, characterized by Cphoto. Bending occurs over a 384 336 B. Phototropism. It was Darwin, at the end of the 19th Cen- characteristic dimensionless bending length `bend := U/Ql 385 337 tury, who demonstrated that exposure of the plant apex to within the tip of the plant. 386 338 a light source was necessary to induce tropic bending (5, 55). 339 Later on, Peter Boysen-Jensen proposed that bending is in- B.1. Fixed light source–no growth. We consider first a fixed light 387 340 duced by a diffusive substance, later identified as auxin, that source and restrict our attention to the case of zero axial 388 341 carries the tropic information from the apex to the rest of growth so that ` = L for all time and the transport equation 389 342 the shoot (56, 57). These early observations are the basis is solved in the reference variables. For a given transport 390 343 of the popular Cholodny-Went model (58, 59) stating that time ttran, the response of the plant is determined by the 391 344 phototropism relies upon three broad mechanisms: (i) sensing characteristic bending length `bend and the response time as 392 345 of light direction at the tip of the shoot; (ii) establishment of a shown in Fig. 5. With small bending length, the response 346 lateral asymmetry of auxin concentration at the tip; and (iii) 347 basipetal transport of this asymmetrical distribution, resulting (A) (C) 348 in differential growth along the shoot (60–62).

349 We model these three steps by considering axial transport 350 of auxin, with an asymmetrical distribution that is established 1.0 1.0 1.0 351 at the shoot apex by the stimulus, treated as a point source of 0.5 0.5 352 light. We suppose that auxin ﬂows basipetally with advective 5 10 5 10 353 velocity U and uptake Q, and assuming slow diﬀusion, the -0.5 354 transport equation is

∂A ∂ (B) (D) 355 − (UA) = −QA. [14] ∂t ∂s bending length

356 where the derivative in space is taken with respect to the

1.0 1.0 357 current arc length s ∈ [0, `]. Additional source/sink terms 0.2

358 can be used to model axial extension without changing the 0.5 0.5 359 evolution of the curvature but are omitted in the first instance. 5 10 5 10 360 We account for the amount and distribution of auxin at each 361 section via a boundary condition at the tip (s = `) and define 362 Atip(x1, x2, t) = A(x1, x2, `, t) that depends on the light source 0.5 response rate 2.5 363 located at p(t) in space, and a scalar I(t) representing its 364 intensity. We then define the unit vector e from the plant tip Fig. 5. Planar phototropic shape evolution for a fixed light source and with no axial 365 to the light source and write it in the plant reference frame: growth, for small and large values of the ratio of response rate to transport rate 366 e(t) = e1d1(`) + e2d2(`) + e3d3(`), as shown in Fig. 3(C). The ttran/tphoto and dimensionless bending length `bend = U/QL. Inset: alignment is characterized by e2(t), such that e2 = 0 when the tip is pointed at the source for 367 line e1x1 + e2x2 in the cross section distinguishes the light this planar case. Further simulation details and parameters provided in SI Section 7. 368 side of the tip from the dark side and defines the asymmetrical 369 distribution of auxin: 393 is localized close to the tip, and the plant is much slower 394 370 Atip(x1, x2, t) = −κI(t)(e1(t)x1 + e2(t)x2) , [15] to orient (comparing (A) and (B)). Increasing the response 395 rate naturally produces a faster orientation and potentially 396 371 where κ characterizes the sensitivity of the phototropic re- an overshoot. If axial auxin transport is much faster than the 397 372 sponse. growth response (ttran tphoto), then the auxin is effectively 398 For constant velocity U, and in the absence of autotropic in steady state at each growth step (as we have assumed for 399 effects, Eqs. (14) and (15)) can be solved exactly (SI Section the cross-sectional transport). This implies that the delay can 400 4), which gives the phototropic curvature law: be neglected and the curvature response at each point depends 401 on the current orientation of the tip. In this case, since the 402 ∂uˆ1 −Q(` − s) ` − s = −C exp e t − [16] response is characterized entirely by the orientation of a single 403 ∂t photo U 2 U point, the motion is very simple: the plant bends to orient 404 ∂uˆ2 −Q(` − s) ` − s with the light, with no oscillations about the state in which 405 = Cphoto exp e1 t − [17] ∂t U U the tip is perfectly oriented with the light (e2 = 0). 406 Contrast this behavior with gravitropism, in which an os- 407 373 where Cphoto := βκI is a single parameter characterizing the cillation about the vertical state is typical unless a strong 408 374 rate of curvature, and from which the phototropic response autotropism response is added. The difference between the 409

6 | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Moulton et al. 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.

(A) (A) (C)

time 1.0 (E) 5 10 (B)

(A)

(B) (D)

0.1 at moment base (D) (B) (C) (C) GT to PT ratio response

sunrise sunrise 0.25 2.5 mass to stiffness ratio

(F) (G) sunset sunset

2.0

Time Time

sunrise sunrise 1.0

Fig. 6. (A) Geometry of the phototropic response stimulated by a lightsource that time follows a circular path of radius R, shifted a distance Y in the transverse ‘horizon’. 1.0 2.0 (B) A full day-night cycle with fast response and long bending length. (C) the addition of autotropic terms enables the plant to return to the vertical during night, when the phototropic signal is absent. Further simulation details and parameters provided in SI Fig. 7. Gravitropism versus phototropism. In (A)-(E), a ﬁxed light source is located Section 7. to the right, at the point (4,1), while gravity points vertically downward. For each parameter set, the tropic response is simulated for the same total time and with equivalent axial growth. The evolving plant shape, deformed under self-weight, is

410 gravitropic law and the phototropic law for fast transport is shown in increasing time from yellow to blue, with the unstressed shape appearing as a dashed line. The moment at the base for the 4 cases is plotted against time in (E). 411 that during gravitropism, each cross section tries to align itself (F) depicts the setup for a plant escaping from the shade under a rigid obstacle. The 412 with gravity, thus creating a conflict at the global level that phototropic signal either points horizontally, if the tip is under the shade, or vertically, 413 results in an oscillatory motion; while during phototropism if the tip is out of the shaded region. (G) a sample simulation showing a successful 414 each cross section tries to align the tip with the light, so there escape. Further simulation details and parameters provided in SI Section 7. 415 is no conflict. However, with delay, such a conflict does exist, 416 due to the fact that each cross section is accessing a previous mechanisms, not considered here, based on circadian rhythms 426 417 state of the tip. Thus, in the regime ttran ∼ tphoto, and if the to reorient the plant at night to face eastward in anticipation 427 418 bending length is not too short, a damped oscillation about of the next sunrise (65). 428 419 the preferred orientation is observed, as shown in Fig.5(C).

B.2. Moving light source. Next, we consider a moving source, and C. Photogravitropic response. Next, we demonstrate the 429 in particular we simulate a day-night cycle of a plant following delicate balance that must exist in the presence of tropic 430 a light source (the sun) as shown in Fig. 6(A). The intensity responses to multiple stimuli. We simulate two different 431 I(t) is also taken to be sinusoidal, so that the phototropic signal scenarios of a plant responding to simultaneous but conflicting 432 is strongest at noon and the signal vanishes at sunset. For fast gravitropic and phototropic signals. Following (12), we 433 response and long bending length, the plant bends significantly assume that the effects of multiple stimuli are additive (see 434 and succesfully tracks the moving light source (Fig. 6(B)). SI Section 4E). This assumption is based on the existence 435 However, at night and without a signal the motion halts (see of separate pathways for signal transduction leading to the 436 also SI movie 5). The plant remains bent towards sunset the redistribution of auxin. However, it is known that these 437 entire night and does not display the nocturnal reorientation pathways share common molecular processes and there are 438 observed in many plants (63, 64). With a non-vanishing non-trivial interactions between different tropisms (66) that 439 autotropic term ξ in Eq. (5), we obtain an autophototropism will not be included here. 440

curvature law of the form 441 ∂uˆ1 −Q(` − s) ` − s = −Cphoto exp e2 t − − ξuˆ1, C.1. Fixed horizontal light source. We consider a growing plant 442 ∂t U U subject to self-weight and initially oriented vertically, but 443 [18] with a ﬁxed light source located in the transverse horizontal 444 ∂uˆ −Q(` − s) ` − s direction. The evolution of the plant can then be characterized 445 2 = C exp e t − − ξuˆ . [19] ∂t photo U 1 U 2 by the ratio of response rates to gravitropic versus phototropic 446 signals, and the ratio of density to bending stiﬀness, which 447

420 The additional terms serve to straighten the plant in the controls the degree of deformation under self-weight. 448 421 absence of any other signal. This is evident in Fig. 6(C), in In Fig. 7(A)-(D) we show the evolving morphology of the 449 422 which we see that the motion during the day is very similar, plant in this two-dimensional parameter space, plotting both 450 423 while at night the stem straightens back to the vertical (see the deformed shape (solid lines) and the reference unstressed 451 424 also SI movie 6). We note that in heliotropic plants such as the shape (dashed lines). In (A), (B), the eﬀect of self-weight 452 425 common sunﬂower, Helianthus annuus, there are additional is relatively minimal, and the evolution is primarily driven 453

Moulton et al. PNAS | July 30, 2020 | vol. XXX | no. XX | 7 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.

454 by the conflicting phototropic signal acting horizontally to plant with the pole results in a change of curvature, a response 514 455 the right, and the vertical gravitropic signal. With increased referred to as thigmotropism. Since initially the plant only 515 456 mass, there is an increased mechanical deformation due to samples a very small region of the pole, the stimulus field is 516 457 self-weight, so that significant disparity develops between the highly localized. As the plant wraps around the pole, new 517 458 deformed and reference shapes. In this regime the balance contact points are established to propagate the helical shape 518 459 of signals has greater importance for the fate of the plant. upward. 519 460 Comparing (B) and (D), the initial phases are similar, but as 461 the plant lengthens and extends to the right in (D), self-weight D.1. Circumnutation. Experiments suggest that, depending on 462 deforms the plant significantly, with half of the plant below the the plant, circumnutation is either driven by an internal os- 463 base level by the end of the simulation. Such a deformation cillator, a time-delay response to gravity, or a combination of 464 could signal failure by creating large torque at the base. In (E) the two (68, 69). Here, following the hypothesis of an internal 465 we plot the moment at the base, where the stress is highest, oscillator, we show that the basic nutating motion emerges 466 against time for each case, and as expected the moment is naturally from an internal oscillator at a single point com- 467 significantly higher for larger mass. bined with axial auxin transport (70). We consider an auxin 468 Intuitively, we expect that this problem could be alleviated source at the point s = sc from which an auxin differential 469 by increasing the gravitropic response rate. Comparing (C) is transported axially. The auxin transport equation may be 470 and (D), the evolution with higher gravitropic response in solved in a similar manner as in the phototropism case with 471 (C) does show decreased sagging. However, the moment at an added rotational component in the local frame of the cross 472 the base is in fact higher in (C). Increasing the gravitropic section due to an internal oscillator (Fig 3(D)). Taking for 473 response rate even further does ultimately alleviate the simplicity a constant rotation rate ω, we obtain (details in SI 474 problem – consider that the plant remains mostly vertical if Section 4) the circumnutation curvature law: 475 gravitropism dominates phototropism – nevertheless, this ex- 476 ample highlights the delicate and potentially counter-intuitive 477 nature of this balance. Q ∂uˆ1 |s − sc| − |s−sc| 478 = C sin ω t − e U , [20] ∂t circ U

479 C.2. Escaping from the shade. The results above suggest a view ∂uˆ2 |s − sc| − Q |s−s | = −C cos ω t − e U c . [21] 480 of a plant as a problem-solving agent that actively responds to ∂t circ U 481 the signals in its environment. A typical problem that many 482 plants have to solve is access to light. For instance, consider

483 a plant growing underneath a canopy as shown in Fig. 7(F) Since the signal here is internal, there is no feedback from the 520 484 and (G). While the tip is in the shaded region, diffuse lighting environment and the morphology of the plant is predetermined 521 485 creates a phototropic stimulus to grow horizontally, orthogonal by the uptake Q, the transport velocity U, the response rate 522 486 to the gravitational signal. If the tip emerges from under the Ccirc, and the rotation rate ω. In Fig. 8(A) we illustrate 523 487 shade, phototropism and gravitropism align, and the plant a sample motion with auxin source at the tip. Fig. 8(B) 524 488 will attempt to grow vertically. In this mixed signal scenario, demonstrates the impact of auxin uptake: high uptake means 525 489 the success or failure of the plant in emerging from under the the motion is constrained to a region very close to the tip 526 490 canopy is down to how the competing signals are integrated, and thus the elliptical shape of the tip pattern is small. More 527 491 and the relative importance of self-weight. An example of a complex tip patterns may also be generated if there is non- 528 492 successful escape is shown in Fig. 7(G). Note that determining uniformity in the internal oscillator (Fig.8(C)). 529 493 the mechanical forces acting in the plant is crucial in this 494 example: once the tip is outside of the shade, both signals 530 495 try to align the entire length of the plant with the vertical, D.2. Thigmotropism. Two interesting observations can be made 531 496 and this leads to physical contact between the plant and the when a twining plant makes first contact with a pole: (i) 532 497 corner of the canopy. Determining the morphology beyond torsion is generated via a localized contact around a single 533 498 this point thus requires determining the mechanical contact point, and (ii) a rotation is induced, i.e. the orientation of the 534 499 force (see SI Section 6) that would not be possible in a purely tangent of the plant with respect to the axis of the pole changes. 535 500 kinematic description. These observations suggest that this contact is sufficient to generate locally a helical shape, and that the pitch of the helix 536 is fixed by internal parameters as opposed to the angle at 537 501 D. Pole dancing. A fascinating plant motion is the mesmeriz- which contact is made (71, 72). 538 502 ing dance that climbing plants, such as twiners, perform to 503 first find a pole and then wrap around it. Like any dance, To show how pole wrapping can be obtained within our 504 this event requires a complex integration of stimuli to achieve framework consistently with these observations, we consider 505 a well-orchestrated sequence of steps: (i) finding a pole, (ii) a plant with a single contact point located at s = 0, and 506 contacting the pole, and (iii) proceeding to wrap around the at position on the boundary Ω0 with angle ψ0 in the plane 507 pole. A common mechanism for searching for a climbing frame d1-d2. The contact induces an auxin gradient at this point, 508 is circumnutation, a combination of circular movement and with maximal auxin on the opposite side of the contact point, 509 axial growth causing the tip to move up in a sweeping spiral i.e. A(0, x1, x2) = −F (cos ψ0 x1 + sin ψ0 x2), and the auxin is 510 path, as first described by Darwin (5, 26, 67). When the transported by an advective flux with both an axial component 511 plant makes contact with a pole, it must then interpret its U and a constant rotational component with angular velocity 512 orientation with respect to the pole in order to wrap around it. ω (Fig 3(E); see also SI Section 4). The transport equation can 513 Here the stimulus is mechanical: the physical contact of the be solved exactly (SI Section 4), and we obtain the following

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(A) (B) (C) 3. Conclusion 564

Plant motion in response to environmental stimuli is a process 565 of extreme biological and ecological relevance. While the pio- 566 neering biologists of the 19th Century investigated the global 567 motion of plants via clever experiments devised to create con- 568 ﬂicting signals and generate complex plant morphologies, most 569 Increasing Increasing of the work of the 20th Century was focused on the molecular 570 and cellular processes, seeking signaling pathways and relevant 571

(D) (E) (F) proteins involved in tropic response. We have combined this 572 accumulated knowledge with recent progress in the physical 573 and computational modeling of living structures to develop a 574 general theory of tropism that relates stimuli to shape. To do 575 so, we created a framework that combines information about 576 auxin transport and the mechanisms by which environmen- 577 tal stimuli are integrated into cellular activities, tissue-level 578 growth, and ultimately a change in shape at the plant scale 579 represented by a morphoelastic structure. 580 Fig. 8. Circumnutation (A)-(C) and thigmotropism (D)-(F). (A) snapshots of a sample circumnutation motion, with tip pattern projected onto the plane. In (B) and (C), tip We have demonstrated the power of this framework through 581 patterns are plotted for varying parameters (the location of the plant base is indicated a series of examples including key effects such as axial growth, 582 by the cross). In (B), an increase in uptake Q decreases the size of the tip pattern. In autotropism, gravitropism, phototropism, thigmotropism, 583 (C), a non-constant angular velocity of the oscillator is given by ω = ωˆ + α cos(5t), self-weight, circumnutation, contact mechanics, and three- 584 generating a tip pattern with 5-fold symmetry, and increasingly non-circular with dimensional deformations. The specific tropic scenarios we 585 increasing α. In (D)-(F), the wrapping around a pole due to thigmotropism via a single contact point is simulated for the same total time, for different parameter regimes: with have considered were chosen to illustrate the range of complex 586 low rotational component (D), the torsion is low, a high rotational component with low behaviors capable of being simulated. The study of individual 587 uptake (E) generates rapid wrapping and high torsion, while wrapping is much slower stimuli provided new laws for the evolution of curvatures for 588 with high uptake (F). Further simulation details and parameters provided in SI Section different form of tropisms. Further, we demonstrated the po- 589 7. tential for multiple and potentially conflicting stimuli to create 590 problems for the plant to solve. The resulting plant behav- 591 iors indicate a need for a delicate balance between competing 592 thigmotropism curvature law: tropisms to achieve a particular task. In each particular sce- 593 nario, we have opted for parsimony over complexity in terms of 594 ˆ ∂u1 Qs ωs modeling choices, in order to highlight qualitative features and 595 = −Cthig exp − sin ψ0 + , [22] ∂t U U how auxin-level differences may become apparent in plant-level 596 ∂uˆ2 Qs ωs morphology. Nevertheless, our approach is fully compatible 597 = Cthig exp − cos ψ0 + . [23] ∂t U U with more detailed modeling choices, though potentially at 598 the cost of increased computation time. 599

539 Solving these equations leads to exact expressions for the intrin- The work presented here integrated information at the tis- 600 540 sic curvatures from which we extract the geometric curvature sue and organ levels. This approach needs to be expanded 601 541 κ = Cthig exp (−Qs/U) t and torsion τ = ω/U (see SI Section to include cell-based and molecular-level descriptions to fully 602 542 4). The curvature increases linearly in time until it reaches link the scales for tropisms and plant growth. At the cell to 603 543 a maximal value determined by the pole radius (the intrinsic tissue scale, the link between auxin and growth in Eq. (5) is 604 544 curvature may keep increasing, but the actual curvature may a lumped description of a complex process that involves cell 605 545 not due to the mechanical contact). For a pole of radius c and wall tension and turgor pressure (73). In principle, additional 606 546 plant radius a the helix radius α = c + a is ﬁxed, while the modeling layers could also be added between the stimulus and 607 547 helical angle φ depends on the rotation rate and is found to auxin response, including for example transcription factors 608 548 satisfy sin(2φ) = ωα/U. and protein production and interactions. Both of these ex- 609

549 For given axial velocity U, the resulting helical shape is de- tensions likely require explicit cell-based modeling. However, 610 550 termined solely by the geometry of the pole and the rotational our framework is such that if the output from a cell model is 611 551 component ω, while the wrapping rate depends on the uptake the value of the piecewise continuous axial growth function 612 552 Q and the response rate Cthig. In Fig. 8(D)-(F) we illustrate g, then the evolution of the curvatures given by Eqs. (7)–(9) 613 553 three different regimes: low rotational component with low still applies and can be used to infer the global changes of 614 554 uptake (D), high rotational component with low uptake (E), geometry. 615 555 and high rotational component with high uptake (F) (see also This work provides a theoretical platform for understand- 616 556 SI movies 7-9). Note that at time κ/ˆ Cthig where κˆ is the fi- ing plant tropisms and generating complex morphologies. As 617 557 nal curvature, the contact point spreads to a contact region, well as linking to cellular and subcellular scales, a key future 618 558 creating a wave of contact and auxin signal that propagates direction is connecting with experiments dedicated to control- 619 559 along the length of the plant. Here, we restrict our attention ling multiple stimuli and generating complex morphologies. 620 560 to the signal from the first contact point. The separate cur- For instance, we showed that a relatively simple experimental 621 561 vature laws for circumnutation and pole wrapping can now set-up like a rotating base under gravity can generate a wide 622 562 be combined to simulate the process of searching for a pole, range of plant shapes. Such steps espouse an approach that is 623 563 making contact, and wrapping (see SI movie 10). both multiscale and multidisciplinary. Indeed, plants refuse 624

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