Downloaded by guest on September 26, 2021 ln tropism plant plant and escape, of canopy twining. to following, integration lead sun as can the such stimuli behavior Further, complex contradictory multiple framework. and/or stimuli pho- this evolving be gravitropic, by can the captured responses three dynamic that easily thigmotropic in examples, and organ nutational, as the totropic, of show, generation deformation We the the to dimensions. and leading forces deformation mechanical tissue of filamen- hormone local in linking explicitly to process by transport stimulus multiscale on arbitrary an this for depending organs model tary stimuli We varying environment. three- with the a the plant in to place growing respect takes with deformation dimensional mechanism feedback orientation overall complex its an This changing by stimuli. hence balanced organ, are the that forces of tissue mechanical create plant deforma- changes the cellular or on local growth or these cell turn, organ stimuli In triggers an tion. these that Typically, of transport stimuli. hormone 2020) movement external 30, induce to directed July review response the for in (received to 2020 organism 4, refers November approved tropism and Kingdom, Plant United Norwich, Center, Innes John Coen, Enrico by Edited a Moulton E. behaviors Derek complex produces tropism in plant stimuli environmental of integration Multiscale omnlrsos sidcd hc ed odfeetcells www.pnas.org/cgi/doi/10.1073/pnas.2016025117 different to leads the which processes. from induced, molecular is ultimately, place response involves, takes hormonal and A cell cue the environmental to The organ an (12). of locations Transduction different multiple signals. in multiple receives of and integration typically an time is plant same movement resulting a exist the there stimulus, at iso- carefully stimuli while particular to Second, one a enable (11). late that stimulus setups weight experimental the branch numerous the change how- increase thus self-weight; may itself and under response breaking growth the of ever, response, likelihood gravitropic a in the vertical the decreasing Similarly, with process. each align may the branch on envi- during tree the a changing impacting so sunflower—is moves, light the sun of of the side grows sunflower intensity it A as influence—the but the varying: ronmental sun, mathe- First, dynamically the face challenge. general be to grand to a grows a tends Yet, is response (10). tropisms growth environment respond of their may description in plants matical change how a and tropisms to of and understanding change, Mars. better the climate as such decide by environments will driven foreign colonizing world conditions, for key a windy be the in may unusually determine rainforests in will of crop must cues future a and environmental of migrate to yield to plants response growth option individual The environment. the of ever-changing lack their Being to plants (9). adapt genomics nature, plant and by biology systems sessile strate- as management well crop as (8), and of gies (7) and biomechanics understanding subject (5) plant our cornerstone a modern for Darwin form of topic Tropisms as central since. such ever a physiology giants been plant (4), has of and contact work (6) to pioneering dates Sachs plants the in response tropisms to of a study back to The thigmotropism, 1). reaction (Fig. others the and many among gravitropism, 3); (1); (2, light gravity to reaction a totropism, P model mathematical ahmtclIsiue nvriyo xod xodO26G ntdKingdom United 6GG, OX2 Oxford Oxford, of University Institute, Mathematical hr,aytoimi udmnal utsaephenomenon. multiscale a fundamentally is tropism any Third, a gaining in role invaluable an plays modeling Mathematical n eomto nrsos osiui ticue pho- includes It stimuli. to growth response directed in of phenomenon deformation general and the is tropism lant | biomechanics a,1,2 are Oliveri Hadrien , | morphoelasticity | o theory rod a,1 n li Goriely Alain and , | doi:10.1073/pnas.2016025117/-/DCSupplemental at online information supporting contains article This 2 1 the under Published Submission.y Direct PNAS a is article This interest.y competing no declare authors The efre eerh ... .. n ..aaye aa n ... .. n A.G. and H.O., D.E.M., and data; A.G. analyzed and A.G. paper. H.O., the and D.E.M., wrote H.O., research; designed D.E.M., A.G. research; and performed H.O., D.E.M., contributions: Author struc- plant even of Moreover, class responses. entire we growth the Here, such to root. undergo or refer may entire stem, that to tropic the vine, tures “plant” branch, a of word single movement the plant, a and use single subset: growth a a the or Within to plant refer tropism. may and of plants response types in variation multiple remarkable the the by complicated further is and (19–22). morphogenesis DNA and from which physiology contexts, rods, to biological elastic proteins multiple of to theory shape applied the been its by has described and be scale, can physical tissue motion with lower-level and endowed filament: the curve by physical space a dictated a as properties by viewed plant in is described the positioning plant well At the and Here, are appropriate. zoom- properties, stimulus is further material external level a shape, the organ field, global or external the plant the level, the in to plant out update the to ing and of properties change, material position global its this the characterize and As To plant (18). time. the stress in of evolve residual shape global generates the also result, it cur- a but creates viewed torsion, growth locally be and nonuniform growth vature can undergoes Differential that (17). plant At remodeling material the 2. and/or of of Fig. continuum cross-section in a depicted each as as level, (1) signaling tissue growth hormone the differential to tissue-level due net expansion a The cell out. is effec- zoom nonuniform the to a assess needs of one direction To effect response, the level. growth to this the relation of at in tiveness stimulus position and environmental its chemical the and plant change of the the the to of understand shape cannot response in one in However, signals. rates molecular different at expanding a,1,2 owo orsodnemyb drse.Eal [email protected] [email protected] Email: addressed. be may correspondence [email protected] whom work.y this To to equally contributed A.G. and H.O., D.E.M., utpesiuisc scnp saeo oewapn for wrapping pole by or escape generated plants. canopy behavior climbing as complex such the stimuli or multiple response stimuli the understand individual feedback to This to updated stimuli. dynamically the be to can respect loop with position plant the drives and modifying shape that thus transport remodeling, and auxin growth into tissue-level them multiple takes transforms that and tropism of stimuli theory changes. level multiscale shape a cellular to propose the leading We to deformations physical down the transduced do create To be to shape. must their signals and changing the sense and light, so, growing to or by ability gravity them their to as respond on such rely signals, plants environmental thrive, multiple to and survive To Significance h hleg ffruaigamteaia oe ftropism of model mathematical a formulating of challenge The y NSlicense.y PNAS . y https://www.pnas.org/lookup/suppl/ NSLts Articles Latest PNAS | f12 of 1

PLANT BIOLOGY APPLIED MATHEMATICS the classical Cosserat rod theory (37–40) to growing filaments. A morphoelastic rod is a one-dimensional filamentary object that can bend and twist with some penalty energy. The rod cannot be elastically stretched, but it can increase in length by addition 3 of mass, leading to a growth stretch. Let r(S, t) ∈ R describe its centerline, where S is the initial arc length measured from the base of the plant toward its tip (Fig. 3A). Together with the fixed Cartesian basis, {ei ; i = 1, 2, 3}, we define, at each point on the curve r(S, t) a local orthonormal basis {di ; i = 1, 2, 3}, oriented such that d3 aligns with the tangent ∂r/∂S in the direc- tion of increasing S, and (d1, d2) denote directions in each cross-section from the centerline to two distinguished material points. From the director basis, the Darboux vector is defined as u = u1d1 + u2d2 + u3d3 and encodes the rod’s curvature, tor- sion, and twist (17). For a given curvature vector, the shape of the rod, and the evolution of the basis, is determined, for boundary conditions {r(0), d1(0), d2(0), d3(0)}, by integrating the system of equations ∂r ∂d = γd , i = γu × d , i = 1, 2, 3. [1] ∂S 3 ∂S i Fig. 1. Classic experiments on tropic responses. (A) Gravitropism: A potted plant realigns itself with gravity (13). (B) Thigmotropism: A twining vine Here γ := ∂s/∂S denotes the total axial growth stretch of each develops curvature when in contact with a pole (6). (C) Phototropism: A section mapping the initial arc length S to the current arc length plant reorients itself toward the light source [18th century experiments by s (41). The general basis specializes to the Frenet–Serret frame Bonnet (14), correctly interpreted by Duhamel du Monceau (15, 16)]. by taking γ = 1 and d1 to be the curve’s normal or to the so-called Bishop frame (or parallel transport frame) by taking γ = 1 and u3 = 0 (42, 43). At each value of S, the cross-section is defined (x , x ) ∈ Ω ⊂ 2 x x within a single plant, multiple environmental cues will combine by a region 1 2 S R , where 1, 2 are local vari- and overlap in effecting mechanotransductive signals, hormonal ables describing the location of material points in the respective d d response, differential growth, and ultimate change in shape directions 1, 2. In terms of the local geometry, any material point X = X1e1 + (23); e.g., a sunflower exhibiting phototropism still perceives a 3 gravitational signal. X2e2 + X3e3 ∈ R in the plant can be represented by its arc At the theoretical level, a variety of approaches have recently length S and its position (x1, x2) on the cross-section at S as been proposed. Growth kinematics models successfully describe follows: the tropic response at the plant level (7, 24, 25), but do not include X = r(S, t) + x1d1(S, t) + x2d2(S, t), for (x1, x2) ∈ ΩS . [2] mechanics and cellular activities. A number of large-deformation elastic rod descriptions of tropic plant growth have also been We can now use this representation to formulate the stimuli. proposed (26–31); these involve a full mechanical description at the plant level, with phenomenological laws for the dynamic updating of intrinsic properties such as bending stiffness and curvature and even branching and self-weight (32), but specific cell- and tissue-level mechanisms are not included. Multiscale formulations have also appeared, including functional–structural plant models (8, 33, 34) and hybrid models with vertex-based cell descriptions (35, 36). These computational approaches have the potential to incorporate effects across scales but are limited to small deformations compared to the ones observed in nature. The goal of this paper is to provide a robust mathematical the- ory that links scales and can easily be adapted to simulate and analyze a large number of overlapping tropisms for a spectrum of plant types. Our mathematical and computational frame- work includes 1) large deformations with changes of curvature and torsion in three-dimensional space; 2) internal and exter- nal mechanical effects such as internal stresses, self-weight, and contact; and 3) tissue-level transport of growth hormone driven by environmental signals. By considering the integration of mul- tiple conflicting signals, we also provide a view of a plant as a problem-solving control system that is actively responding to its environment. 1. Multiscale Modeling Framework The key to our multiscale approach is to join three differ- ent scales: stimulus-driven auxin transport at the cellular level, tissue-level growth mechanics, and organ-level rod mechanics. Fig. 2. Tropism is a multiscale dynamic process: The stimulus takes place at the plant or organ level and its information is transduced down to the A. Geometric Description of the Plant. We start at the organ scale cellular level, creating a tissue response through shape-inducing mechanical and model the plant as a growing, inextensible, unshearable elas- forces that change the shape of the organ. In the process, the plant reorients tic rod following the formalism of ref. 17, chap. 5 that extends itself and, accordingly, the stimulus changes dynamically.

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( (4). helically transported attenuation turn signal in is ( which (E contact (2), a torsion. file by transported and is triggered profile is curvature apical wrapping pulsation The generating (1). with (2), apex oscillator the basipetally at internal profile profile auxin an auxin rotating ( by apical with attenuation generated an with of is basipetally establishment tation transported the is in that results (1) and apex plant (C the (2). flow auxin ( lateral vector causes gravity The Gravitropism: (B) s olo tal. et Moulton 3. Fig. oine coeal)adeupe ihalclmtra basis material local a with equipped and acropetally) (oriented = 3 hscly tml r ed cigi pc tapoint a at space in acting fields are stimuli Physically, n hnigoe time over changing and f (X, ahcross-section Each (A) t .. hmcl eprtr,o ih nest;vec- intensity; light or temperature, chemical, e.g., ), F = F(X, rpcsiuiaecaatrzdb hi origin, their by characterized are stimuli Tropic t .. emgei ed rvt,o light or gravity, field, geomagnetic e.g., ), Ω s httoim h ih etri esdat sensed is vector light The Phototropism: ) fterdi aaeeie yisaclength arc its by parameterized is rod the of t hycnb ihrsaa fields, scalar either be can They . lctn naymtia ui pro- auxin asymmetrical an eliciting 1) ssne nec rs-eto and cross-section each in sensed is 1) oters ftepatwith plant the of rest the to 3) h circumnu- The (D) 2). hgorpcpole Thigmotropic ) ω {d associated 1 , d 2 , X d 3 ∈ }. t 4)it igeojc eciigtelclcag fshape of change local the describing activ- object genetic single and a metabolic tissue into cell (47) and the ity properties, via to material regulated con- all cell due multiple geometry, pressure, deformation the local the integrates of field dictating tributions tensor tensor this growth Physically, a growth. point plant each at the assigns that of (17) concentration morphoelasticity the of to theory level general tissue the Eq. at of field solution the from known is Remodeling. tion and Growth Tissue-Level D. n dition Eq. transport, in auxin limit The either diffusion term. the contained zero about is information the the stimuli tropism, to and particular attention the dominated, on our advection sec- Depending is restrict Appendix, process we (SI the so that velocity suggests and 4A) diffusion tion component auxin diffusive estimated a with component of stimulus sum a a is flux and The turnover, auxin of rate the basis: local stimulus the vector in in a stimulus of the light in case decompose the plant to therefore In the must source. response of the light orientation cellular the of relative to the the relation position must to example, the linked we is For and space, phototropism orientation space. in the points in account at plant physical into defined in grow is take plants stimulus the- also since physical a appropriate a Since is tropism fields). (which of plant fields makes functions of which general ory time in and are stress— space stimuli mechanical both These (e.g., here). fields considered tensor not possibly, and, direction; where quantities The o h ui concentration (58) auxin con- equation modeled the for are reaction–advection–diffusion as effects standard such These a mechanisms (55–57). through various oxidation direct by or removed jugation locally and signal concentration. tion auxin tropic on of only function depend a to taken is is flux growth auxin and (54). model organism understood our completely the in not at Therefore, are curvature that global pathways through elicits asym- level resulting cells auxin The of of growth. levels growth faster higher metrical with shoots, associated In in generally (53). transporters are signals auxin tropic of relocalization to the response gra- via lateral controlled A is 52). dient a (51, lat- as tropisms Indeed, accepted underlying remodeling. broadly mechanism to universal is and redistribution growth known auxin plant is asymmetrical in which erally line role auxin central in phytohormone a is the play This 50). consider morphogen. ref. we (e.g., a growth Here, of morphogen-mediated of role concen- models anelastic the hormone other the with plays single which that a in field via view tration locally simplic- coarse-grained connected for a are hence, they level; here expansions (47–49), continuum adopt the available we to ity are extend processes easily not cellular do these detailed While of growth. to as refer models shape We cellular 46). of (45, change geometric tension any turgor-induced cell to the response of in expansions Transport. walls anelastic through Auxin place take and deformations Response Cellular C. cellular the to stimulus external an link response. we Next, plant. the by felt sa uwr omlvco oteboundary the to vector normal outward an is easm htaxni rnpre ydfuinadadvec- and diffusion by transported is auxin that assume We J F J · ∂ = ∂ n steaxnflux, auxin the is A t 0 = F + 1 d · ∇ tteotrbudr fec rs-eto,where cross-section, each of boundary outer the at (F 1 + 1 J , F F = 2 2 d −QA , 2 F J + 3 stim ) F r h opnnso h tmlsas stimulus the of components the are Q + lhuhasml cln analysis scaling simple a although , A(x 3 d C saprmtrta characterizes that parameter a is 3 scmie ihan-u con- no-flux a with combined is 4, J , C 1 stim , , F x atrsaysucso sinks. or sources any captures 2 i ri onayo source or boundary a in or , = S ecnrlt h growth the relate can we 4, J , F neteaxndistribu- auxin the Once = NSLts Articles Latest PNAS t · ): −κ∇ d tteclua level, cellular the At i , ee euethe use we Here, A. J A ∂ i diff Ω ,2 3 2, 1, = + = S J . −κ∇A stim , | . κ f12 of 3 we F, → and [3] [4] 0.

PLANT BIOLOGY APPLIED MATHEMATICS of an elementary volume element (59). This growth tensor may Since at this scale the plant is treated as a one-dimensional be different in different directions (anisotropic growth) and/or structure, its equilibrium shape is easily computed even in com- spatially varying (heterogeneous growth) (60) to encode changes plex nonplanar geometries. Once the deformation is determined, in both length and girth. However, here we assume that growth the multiscale cycle is completed by updating the map between takes place, locally, only along the axial direction and not in the external stimulus and cell-scale response with respect to the cross-sectional direction. This assumption implies that there is updated orientation, and the process is repeated. no change in thickness, an effect that could be of importance in some systems. Then, the only nontrivial component of the growth G. Summary. The flow of information between different spatial tensor is a single function g(x1, x2, S, t) that describes the change scales for a given stimulus field proceeds as follows: 1) Given of axial length of an infinitesimal volume element (SI Appendix, an initial plant shape, a stimulus F impacts auxin transport and section 2). An initially straight filament of length L0 with g con- thus local concentrations of auxin via the transport equation (Eq. stant at all points would grow to a new straight filament of length 4); 2) the local auxin concentration A changes the local growth L = L0(1 + g(t)) (18). If, however, g = g(x1, x2, S, t) varies from field that impacts the intrinsic curvatures and axial extension of point to point, the same filament would tend to bend and twist as the plant via Eqs. 6–9; and 3) the new intrinsic curvatures and shown in Fig. 2. external conditions determine the new mechanical equilibrium Next, we connect the axial growth function g = g(x1, x2, S, t) of the plant, thus changing the plant position and shape in the to the concentration of auxin A = A(x1, x2, S, t) via a growth law stimulus field. of the form The theoretical objective in this work is to bridge the divide ∂g = β(A − A∗) − ξ(g − g), [5] between cell-based descriptions of auxin transport and plant-level ∂t descriptions of tropism kinematics. In the examples below, we where A∗ is a baseline level of auxin, β characterizes the rate at demonstrate how the tissue-level transport and growth equations which an increase in auxin generates growth, and may be mathematically combined to yield explicit evolution rules for the curvature and axial growth at the rod level. In this way, 1 Z g = g dx dx the multiscale flow can be efficiently simulated and analyzed for a A 1 2 ΩS variety of tropic responses. It is also worth noting that a significant amount of biology exists between the cell-scale and the tissue- is the average of the growth field, with A the area of the cross- level models we propose, therefore we largely opt for qualita- section. The term ξ(g − g) provides a pointwise measure of the tive investigation of complex behavior, with further experimental strain induced by differential growth and models autotropism, and theoretical work needed to refine parameter selection. the observed tendency to grow straight when subject to other tropisms. The underlying mechanisms of autotropism are poorly 2. Examples understood, but studies using radiolabeled auxin suggest that this A. Gravitropism. Gravitropism has been extensively studied both straightening response does not depend on auxin but rather is experimentally and theoretically. The classic description is based sensed via an actomyosin-dependent mechanism (61, 62). on the so-called “sine rule” in which the change in curvature fol- lows the sine of the angle with the direction of gravity (64). While E. Change in Local Shape and Properties. The axial growth func- it is successful in capturing observed behavior in gravitropic tion g is defined at the tissue scale and, as such, does not directly experiments, it is mostly phenomenological and is applicable only give the change in curvature and torsion of the plant. Indeed, the to planar geometry. Here, we show that the sine rule emerges change of shape depends not only on g but also on the internal naturally from our formulation but that it can be generalized to mechanical balance of the forces generated by each growing vol- include three-dimensional deformations that are generated when ume element. Following the general theory given in ref. 18 and the entire plant is forced to change its orientation in time. its adaptation to the particular case of plants given by the growth The stimulus for gravitropism is the vector field F = −Ge3 law Eq. 5, we compute the intrinsic curvatures and elongation which can be written in the plant frame of reference as F = f + of the growing plant (SI Appendix, section 3). In the absence f3d3, where f := f1d1 + f2d2 is the gravity force acting in the plane of autotropism (ξ = 0), these curvatures (given by the vector uˆ), of the cross-section. Since it is believed that plants are insensi- which define the shape of the plant in the absence of body force tive to the strength of gravitational field (65), it is sufficient to and external loads, evolve via use a unit vector representing only the direction of gravity; i.e., ∂uˆ Z we scale the gravitational acceleration G to 1. If f ≡ 0, no tropic I 1 = β x A dx dx , [6] response will occur. Gravity perception relies on specific cells ∂t 2 1 2 ΩS called statocytes distributed along the shoot (66). Statocytes con- Z ∂uˆ2 tain dense organelles, statoliths, that sediment under the effect I = −β x1A dx1dx2, [7] of gravity. Tilting of the plant causes statoliths to avalanche and ∂t Ω S to form a free surface perpendicular to the gravity vector, pro- ∂uˆ 3 = 0, [8] viding orientational information to the cell (67). It has been ∂t observed that the gravitropic response depends upon the angle Z ∂γ ∗ between the statoliths free surface and the vertical, but not upon A = β (A − A ) dx1dx2. [9] ∂t Ω the intensity of the gravitational field or the pressure of sta- S toliths against the cell membrane (65). A possible mechanism Here and for the rest of this paper, we have assumed that the is that the contact between the statoliths and the cell membrane cross-section is circular with radius R, area A = πR2, and second may trigger relocalization of PIN membrane transporters and a moment of area I = πR4/4. redirection of auxin flux (67). Here, we follow this hypothesis and, accordingly, assume that gravity drives an advective flow F. Rod Mechanics Set the Plant Position in the Stimulus Field. Once of auxin Jstim = kAf. If the statocytes are uniformly distributed the intrinsic curvatures and elongation of the plant following within the stem volume, then k is constant. We assume also a growth have been updated, the plant position and orientation are source and sink of auxin on each cross-section, representing a updated by solving the Kirchhoff equations (41, 63) for the bal- continual axial auxin flow, and that auxin transport occurs on ance of linear and angular momentum for given external forces timescales much shorter than the one associated with growth. such as self-weight, wind, or contact forces (see SI Appendix, Combining the transport equation (Eq. 4), the growth law section 1 for details on the Kirchhoff equations). (Eq. 5) (in the absence of autotropism for simplicity), and the

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PLANT BIOLOGY APPLIED MATHEMATICS Boysen-Jensen proposed that bending is induced by a diffu- auxin is effectively in steady state at each growth step (as we have sive substance, later identified as auxin, that carries the tropic assumed for the cross-sectional transport). This implies that the information from the apex to the rest of the shoot (73, 74). delay can be neglected and the curvature response at each point These early observations are the basis of the popular Cholodny– depends on the current orientation of the tip. In this case, since Went model (75, 76) stating that phototropism relies upon three the response is characterized entirely by the orientation of a sin- broad mechanisms: 1) sensing of light direction at the tip of gle point, the motion is very simple: The plant bends to orient the shoot; 2) establishment of a lateral asymmetry of auxin with the light, with no oscillations about the state in which the tip concentration at the tip; and 3) basipetal transport of this asym- is perfectly oriented with the light (e2 = 0). metrical distribution, resulting in differential growth along the Contrast this behavior with gravitropism, in which an oscilla- shoot (77–79). tion about the vertical state is typical unless a strong autotropism We model these three steps by considering axial transport of response is added. The difference between the gravitropic model auxin, with an asymmetrical distribution that is established at the and the phototropic model for fast transport is that during grav- shoot apex by the stimulus, treated as a point source of light. itropism, each cross-section tries to align itself with gravity, thus We suppose that auxin flows basipetally with advective velocity creating a conflict at the global level that results in an oscilla- U and turnover Q, for which the transport equation is tory motion; while during phototropism each cross-section tries to align the tip with the light, so there is no conflict. However, ∂A ∂ − (UA)= −QA. [13] with delay, such a conflict does exist, due to the fact that each ∂t ∂s cross-section is accessing a previous state of the tip. Thus, in the regime ttran ∼ tphoto, and if the bending length is not too short, a Here the derivative in space is taken with respect to the current damped oscillation about the preferred orientation is observed, arc length s ∈ [0, `]. Additional source/sink terms can be used to as shown in Fig. 5C. model axial extension without changing the evolution of the cur- vature (SI Appendix, section 4C). We account for the amount and B.2. Moving light source. Next, we consider a moving source, and distribution of auxin at each section via a boundary condition in particular we simulate a day–night cycle of a plant follow- at the tip (s = `) and define Atip(x1, x2, t) = A(x1, x2, `, t) that ing a light source (the Sun) as shown in Fig. 6A. The intensity depends on the light source located at p(t) in space and a scalar I (t) is also taken to be sinusoidal, so that the phototropic sig- I (t) representing its intensity. We then define the unit vector e nal is strongest at noon and the signal vanishes at sunset. For from the plant tip to the light source and write it in the plant fast response and long bending length, the plant bends signifi- reference frame e(t) = e1(t)d1(`)+ e2(t)d2(`)+ e3(t)d3(`), as cantly and successfully tracks the moving light source (Fig. 6B). shown in Fig. 3C. The vector e1d1 + e2d2 in the cross-section dis- However, at night and without a signal, the motion halts (Movie tinguishes the light side of the tip from the dark side and defines S5). The plant remains bent toward sunset the entire night and the asymmetrical distribution of auxin: does not display the nocturnal reorientation observed in many plants (80, 81). With a nonvanishing autotropic term ξ in Eq. 5, Atip(x1, x2, t) = −κI (t)(e1(t)x1 + e2(t)x2), [14] we obtain an autophototropism curvature model of the form

where κ characterizes the sensitivity of the phototropic response.     For constant velocity U , and in the absence of autotropic ∂uˆ1 −Q(` − s) ` − s = −Cphoto exp e2 t − − ξuˆ1, [17] effects, Eqs. 13 and 14 can be solved exactly (SI Appendix, section ∂t U U     4C), leading to the phototropic curvature model: ∂uˆ2 −Q(` − s) ` − s = Cphoto exp e1 t − − ξuˆ2. [18] ∂uˆ  −Q(` − s)   ` − s  ∂t U U 1 = −C exp e t − [15] ∂t photo U 2 U ∂uˆ  −Q(` − s)   ` − s  2 = C exp e t − , [16] A C ∂t photo U 1 U

where Cphoto := βκI is a single parameter from which the pho- 1.0 1.0 totropic response time is defined as tphoto := 1/(LCphoto). The 1.0

exponential decay in Eq. 15 is due to the turnover of auxin so that 0.5 0.5

less is available at the base, while the time shift ttran := L/U of 5 10 5 10 Atip accounts for the transport time to the section at arc length s, -0.5 leading to time-delay differential equations. The change of curva- ture thus depends on three quantities: 1) the orientation of the tip with respect to the light source t = (` − s)/U ago; 2) the amount B D of auxin available for the phototropic signal, which depends on the turnover Q; and 3) the plant’s response sensitivity, charac- bending length terized by Cphoto. Bending occurs over a characteristic dimension- 0.2 1.0 1.0 less bending length `bend := U /(Q`) within the tip of the plant. B.1. Fixed light source—no growth. We consider first a fixed light 0.5 0.5 source and restrict our attention to the case of zero axial growth so that ` = L for all time and the transport equation is solved 5 10 5 10 in the reference variables. For a given transport time ttran, the response of the plant is determined by the characteristic dimen- 0.5 response rate 2.5 sionless bending length `bend and the response time as shown in Fig. 5. Fig. 5. (A–D) Planar phototropic shape evolution for a fixed light source With small bending length, the response is localized close to and with no axial growth, for small and large values of the ratio of response the tip, and the plant is much slower to orient (comparing Fig. 5 rate to transport rate ttran/tphoto and dimensionless bending length `bend = A and B). Increasing the response rate naturally produces a faster U/(QL). (A–D, Insets) Alignment is characterized by e2(t), such that e2 = 0 orientation and potentially an overshoot. If axial auxin transport when the tip is pointed at the source for this planar case. Further simulation is much faster than the growth response (ttran  tphoto), then the details and parameters are provided in SI Appendix, section 8.

6 of 12 | www.pnas.org/cgi/doi/10.1073/pnas.2016025117 Moulton et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 olo tal. et Moulton efwih eom h plant the deforms self-weight 7D, and lengthens Fig. plant in the 7 right as Fig. but the greater Comparing similar, to has are plant. extends signals phases the initial of of the balance fate D, the the regime for refer- this importance and that In deformed so shapes. the self-weight, ence between an to develops is disparity due there significant mass, deformation increased mechanical With increased signal. the and gravitropic by right vertical the driven to the primarily horizontally acting is signal evolution phototropic conflicting the and 7 minimal, Fig. relatively shape In unstressed lines). reference the (dashed and the lines) both (solid plotting shape space, deformed parameter two-dimensional this in plant the controls which stiffness, self-weight. bending under to deformation signals of density phototropic degree of versus ratio the gravitropic direc- the by to characterized and horizontal rates be then response transverse can of the plant ratio the with in of but located evolution The vertically, source tion. oriented light initially fixed and a self-weight to source. subject light horizontal here. Fixed included C.1. not molecular are different common that between (83) share interactions tropisms nontrivial pathways are these there How- that and for auxin. processes known pathways of is redistribution separate it the of ever, This to existence ). 4F leading effects the transduction section the on signal Appendix, that based (SI is assume additive assumption are we stimuli 12, multiple ref. of pho- Following to and plant signals. gravitropic responses a conflicting totropic of tropic but scenarios of simultaneous different to presence two responding simulate the We in stimuli. exist multiple must that balance Response. Photogravitropic C. face (82). to sunrise night next at the plant of the anticipation reorient in here, eastward to considered rhythms not circadian mechanisms, on that sunflower, based additional note common are We the there S6). see as the annuus, (Movie we such night vertical plants at which while heliotropic the in similar, in to 6C, very back Fig. is straightens day in the stem evident during is motion This the absence the that signal. in other plant the any straighten of to serve terms additional The is signal in phototropic provided the 8. parameters section when and night, details during simulation vertical Further absent. the and to response return fast to (C with plant cycle radius length. day–night of bending full A path long (B) circular “horizon.” transverse a the follows that source 6. Fig. BC

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sunrise hfe distance a shifted R, IAppendix, SI Helianthus B Y and in h oa rm ftecosscindet nitra oscilla- model: internal rate curvature rotation an in constant to a (details simplicity due obtain for cross-section in Taking 3D). component the (Fig rotational of tor in added frame that to an local manner with the similar case a phototropism in solved the auxin be The point may axially. auxin the equation transported is at axial transport differential source with auxin auxin an combined which an from point consider an We single from (89). a naturally transport emerges at we motion oscillator oscillator, nutating internal internal basic an (87, the of two a that hypothesis the show oscillator, the of internal combination following a Here, an or 88). by gravity, to driven response is time-delay circumnutation plant, the Circumnutation. pole, D.1. stimu- shape the the helical around the pole, propagate wraps upward. to plant the established the are of points As contact region localized. new highly small curva- is of very field the change a lus initially a Since only in thigmotropism. samples as results physical plant to pole referred The the response mechanical: with a is ture, plant must stimulus to the it order the of pole, in Here contact a pole it. the with to around contact respect Darwin wrap with makes by orientation plant described its the first interpret then as When path, to 86). spiral tip 26, sweeping the (5, combi- causing a a growth in axial circumnutation, up and move is movement frame mechanism circular common climbing of A and nation a pole. pole, for the the searching contacting around for 2) wrap pole, well-orchestrated to a proceeding a finding 3) achieve 1) to steps: stimuli of requires event sequence of this integration dance, any complex Like a it. a find around first wrap to then perform and twiners, pole as such plants, climbing that dance Dancing. Pole D. possible be not would description. kinematic which purely 6), a section in , Appendix con- (SI mechanical the force determining tact the requires morphology thus between the point Determining contact this canopy. plant physical beyond the the of to corner of of leads the length outside and this entire plant is and the tip vertical, align the to the Once try with example: signals acting this both forces in shade, mechanical crucial the the is plant determining shown the that is in escape Note successful 7G. a Fig. of in example impor- An relative the self-weight. and to of down integrated or tance is are canopy success signals the competing the under the from scenario, how emerging in mixed-signal plant the this of attempt In failure will vertically. plant grow the shade, and the to align, under from gravitropism to emerges and orthogonal tip phototropism the horizontally, 7 If grow signal. a Fig. to gravitational consider the in stimulus instance, phototropic shown a For as ates (85) (84). canopy light a to plants and underneath many access growing that is problem plant the solve typical to to A responds environment. have actively that its agent in problem-solving signals a as plant a shade. the from balance. Escaping this C.2. of nature counterintuitive potentially dominates and delicate gravitropism the problem—consider if highlights example vertical the this mostly phototropism—nevertheless, rate remains alleviate response plant ultimately gravitropic the that the does in Increasing further is 7C. base even Fig. the in at moment higher the fact 7C However, Fig. in sagging. response decreased gravitropic show higher with 7 evolution Fig. the Comparing D, rate. response gravitropic the increasing for higher 7 significantly for is time Fig. moment against mass. the In larger highest, expected is as base. stress and the case, the failure each where base, at the signal the by could torque at level deformation moment large base a creating the Such below simulation. by plant the the of of end half with significantly, nutvl,w xetta hspolmcudb leitdby alleviated be could problem this that expect we Intuitively, hl h i si h hddrgo,dfuelgtn cre- lighting diffuse region, shaded the in is tip the While G. acntn ln oini h mesmerizing the is motion plant fascinating A h circumnutation the 4D) section Appendix, SI xeiet ugs ht eedn on depending that, suggest Experiments h eut bv ugs iwof view a suggest above results The NSLts Articles Latest PNAS E epo the plot we | C s ω f12 of 7 does = we , and s F c

PLANT BIOLOGY APPLIED MATHEMATICS A C

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(A)

B D

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0.25 2.5 mass to stiffness ratio

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Fig. 7. Gravitropism versus phototropism. In A–E, a fixed light source is located 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 shown in increasing time from yellow to blue, with the unstressed shape appearing as a dashed line. The moment at the base for the four cases is plotted against time in E. F depicts the setup for a plant escaping from the shade under a rigid obstacle. The phototropic signal points either horizontally, if the tip is under the shade, or vertically, if the tip is out of the shaded region. (G) A sample simulation showing a successful escape. Further simulation details and parameters are provided in SI Appendix, section 8.

ˆ    Q ∂u1 |s − sc | − |s−sc | may also be generated if there is nonuniformity in the internal = Ccirc sin ω t − e U , [19] ∂t U oscillator (Fig. 8C).    D.2. Thigmotropism. Two interesting observations can be made ∂uˆ2 |s − sc | − Q |s−s | = −C cos ω t − e U c . [20] ∂t circ U when a twining plant makes first contact with a pole: 1) Torsion is generated via a localized contact around a single point, and 2) Since the signal here is internal, there is no feedback from the a rotation is induced; i.e., the orientation of the tangent of the environment and the morphology of the plant is predetermined plant with respect to the axis of the pole changes. These obser- by the turnover Q, the transport velocity U , the response rate vations suggest that this contact is sufficient to generate locally Ccirc, and the rotation rate ω. In Fig. 8A we illustrate a sample a helical shape and that the pitch of the helix is fixed by inter- motion with auxin source at the tip. Fig. 8B demonstrates the nal parameters as opposed to the angle at which contact is made impact of auxin turnover: High turnover means the motion is (90, 91). constrained to a region very close to the tip and thus the ellip- To show how pole wrapping can be obtained within our frame- tical shape of the tip pattern is small. More complex tip patterns work consistently with these observations, we consider a plant

8 of 12 | www.pnas.org/cgi/doi/10.1073/pnas.2016025117 Moulton et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 E ( turnover low with al. component et rotational Moulton high a (F low, turnover is high torsion with the slower much (D) is component wrapping rotational while low torsion, in a high and With wrapping regimes: rapid parameter generates different for time, total increasing with noncircular increasingly and h ieo h i atr.In pattern. tip the of size the In curvature thigmotropism following the sec- Appendix, obtain model: (SI we exactly and solved 4E), be tion velocity component can angular equation axial transport with The an component i.e., both rotational point, stant with contact flux the advective auxin of maximal side with −F point, opposite this the at gradient on auxin an induces tact boundary the i.8. Fig. radius of pole may a curvature For intrinsic actual contact). radius (the mechanical the plant the radius but to pole due increasing, the not keep by may reaches determined it curvature until value time maximal in curvature linearly a increases geometric curvature The the 8F). tion extract we which intrin- κ the from for expressions curvatures exact sic to leads equations these Solving at located point contact single a with = B . 8 section Appendix , SI DE C AB A and (cos C thig (A–F i atrsaepotdfrvrigprmtr telcto ftepatbs sidctdb h rs) In cross). the by indicated is base plant the of location (the parameters varying for plotted are patterns tip C, ψ x (−Qs exp 0 icmuain(A–C Circumnutation ) ∂ ∂ x ∂ ∂ 1 a u u ˆ ˆ t t sin + 1 2 Ω h ei radius helix the = = 0 C −C /U ihangle with thig ψ 0 thig )t exp x 2 exp ocntn nua eoiyo h siltri ie by given is oscillator the of velocity angular nonconstant a C, n torsion and n h ui stasotdb an by transported is auxin the and ),  −  n hgorps (D–F thigmotropism and ) − ψ Qs U α 0 Qs U =  nteplane the in c cos  τ + sin In α. =  s a U ω/ ψ 0 =  sfie,wieteheli- the while fixed, is h rpigaon oedet hgorps tasnl otc on ssmltdfrtesame the for simulated is point contact single a at thigmotropism to due pole a around wrapping the D–F, 0 ψ Increasing + 0 n tpsto on position at and + sec- Appendix, (SI ω U d s ω U 1 U  s -d A(0, .  npht fasml icmuainmto,wt i atr rjce noteplane. the onto projected pattern tip with motion, circumnutation sample a of Snapshots (A) ). ω 2 n con- a and , h con- The . Fg 3E). (Fig. x 1 , c x [21] [22] 2 and = ) sin(2 angle cal uptwt aai he itnteprmna cnro that scenarios experimental distinct three in of model data compare aim we with 9 the Fig. validate in output with construct, To general behavior. a rather as complex framework but of the range experiments, diverse a or demonstrating plants focused specific not were far on thus considered have we scenarios tropic The Validation Model wrapping 3. and the contact, simulate making to pole, combined a be for ( now searching circumnuta- can of for wrapping process models first pole the curvature and from separate signal of tion the The the wave to point. of attention a length our contact the creating restrict along we region, propagates Here, that contact plant. signal a auxin and to contact spreads point time contact (F at turnover that high Note ( with turnover component low turnover with rotational low component with rotational component high rotational (D), rotational the low and regimes: different pole the of Q geometry the component by solely mined oi S10). Movie o ie xa velocity axial given For n h epnerate response the and = φ) ω ˆ = φ U ωα/ ω eed ntertto aeadi on osatisfy to found is and rate rotation the on depends ω + hl h rpigrt eed nteturnover the on depends rate wrapping the while , α .Frhrsmlto eal n aaeesaeprovided are parameters and details simulation Further ). Increasing . cos(5t κ/C ˆ F ,gnrtn i atr ihfieodsymmetry, fivefold with pattern tip a generating ), thig C U thig where h eutn eia hp sdeter- is shape helical resulting the , nFg 8 Fig. In . nices nturnover in increase an B, κ ˆ stefia uvtr,the curvature, final the is NSLts Articles Latest PNAS D–F oisS7–S9). (Movies ) eilsrt three illustrate we ,adhigh and E), Q decreases | f12 of 9 )

PLANT BIOLOGY APPLIED MATHEMATICS together include all of the tropisms we have modeled: Fig. 9A tions, demonstrating a robustness of the framework across a range includes data on thigmotropic curvature generation (92) and grav- of plant types and combined tropic responses. itropic bending, Fig. 9B shows a diversity of tip patterns mea- A second type of validation is obtained by considering the sim- sured during circumnutation (93), and Fig. 9C plots the evolving plification of our models to existing purely kinematic models. shape of saplings bending to align with gravity and exposed to A number of geometric models exist in the literature, positing isotropic (Fig. 9 C, Left) or anisotropic (Fig. 9 C, Right) light (94) the evolution of the plant’s curvature as a function of time, and (see SI Appendix, section 9 for details of these experiments and have been systematically validated against data and observations. model comparison). In each case, the model is able to repro- The sine law (Eq. 12) is an example of such a kinematic model. duce, qualitatively and quantitatively, the experimental observa- Similarly, planar kinematic phototropic (12) and circumnutation

A

Thig + data

Grav (degrees) model + Auto

Time (h)

B

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80

C 60

60 Photo (symbols) data + 40 model Grav + 40 Auto y-coordinate (cm) y-coordinate (cm) 20 20

Light source 10 20 30 10 20 30 x-coordinate (cm) x-coordinate (cm)

Fig. 9. A comparison of model output with experimental observations for a variety of tropic responses. (A) Mechanical perturbation is applied to cucumber hypocotyls situated vertically, causing them to bend (blue data and curves), after which they recover the vertical; a horizontally situated plant bends toward the vertical under gravitropism (red data and curves). (B) Multiple tip patterns are observed in sunflowers; these circumnutation patterns are reproduced by the model with (solid curves) and without (dashed curves) gravitropic effects. (C) Tree saplings are inclined at an angle and subjected to either isotropic (Left) or anisotropic (Right) light. The shape of the plants is extracted at five times and discretized along the length (symbols). Continuous two-dimensional shapes (solid curves) obtained by our model combining gravitropic, phototropic, and autotropic effects are included at the same time points.

10 of 12 | www.pnas.org/cgi/doi/10.1073/pnas.2016025117 Moulton et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 4 .Bonnet, C. 14. Photo-, plants: in tropism Pfeffer, P. shoot F. of W. model unified 13. A Moulia, root B. Douady, into S. Bastien, insights R. New 12. Swarup, R. Mattheck, Vissenberg, C. Mattheck, K. M. C. Bennett, 11. M. Hijazi, H. Sato, E. 10. olo tal. et Moulton n ra ees hsapoc ed ob xaddt include to expanded be to needs approach This levels. organ and plant-level in and apparent become features morphology. in may qualitative complexity differences highlight over auxin-level to how parsimony choices, for modeling partic- opted each of In terms have between task. we balance particular scenario, delicate a ular a achieve for to tropisms need competing resulting a The indicate solve. to behaviors plant the plant for problems conflict- create potentially to and stimuli we ing multiple Further, for needed. potential as the refined, experiments demonstrated and and confronted data be against can iteratively, models These different stimuli for tropisms. individual curvatures of of of have forms evolution study the for we The models simulated. new provided scenarios being behav- complex of of tropic capable range the iors illustrate specific to chosen three-dimen- were The considered and deformations. mechanics, growth, sional contact axial self- as circumnutation, thigmotropism, such weight, effects phototropism, key gravitropism, including autotropism, examples of series at a shape structure. in morphoelastic activities, change a as a cellular viewed scale to into plant which ultimately the by leading integrated mechanisms growth, are the tissue-level do and To stimuli transport shape. to auxin environmental gen- stimuli modeled a relates we develop that so, tropism to for structures this framework living eral combined and of physical have the modeling in We computational progress molecular recent response. with the tropic knowledge relevant on accumulated in and focused pathways involved was signaling century proteins seeking 20th processes, the cellular most con- of and morphologies, create work plant global to the complex devised of the generate experiments and investigated clever signals century via flicting pio- 19th plants the of the While motion process of relevance. a biologists ecological is and neering stimuli biological environmental extreme to of response in motion Plant Conclusion 4. organ-level in seen level be the might growth at kinematics. tissue changes and how transport into auxin insight of potential enables descrip- and/or and these generalizes tions restrictions both geometric framework Our in particular limits. parameter under explicitly derived show have we we and proposed 7 section been Appendix, have (95) models .C so,U eml,E icm ln rpss rvdn h oe fmovement of power the Providing tropisms: Plant Liscum, E. Pedmale, U. Esmon, C. 9. A plants: Vos in movements J. gravitropic of 8. control and power The Fournier, M. Moulia, B. 7. Sachs, J. 6. Darwin, C. 5. S D. Simmons, C. 4. reveals gravitropism shoot of model Unifying Douady, S. Moulia, B. Bohr, T. Bastien, R. 3. Band R. L. 2. .R .Fr,J ib,Teetbiheto rpccraue nplants. in curvatures tropic of establishment The Digby, J. Firn, D. R. 1. h okpeetdhr nertdifraina h tissue the at information integrated here presented work The through framework this of power the demonstrated have We lIpiei eSme ace iried o,1779). Roi, du Libraire Fauche, Samuel de (l’Imprimerie organism. sessile a to science. view. biology systems and biomechanical 1888). thaliana. Arabidopsis in plants. in control posture of U.S.A. feature central a as proprioception (2012). mechanism. tipping-point a by controlled rfwcsl ndrPflanze der in Kraftwechsels propio-ception. and gravi- 1998). Media, Business & signalling. gravitropic Physiol. 5–6 (2013). 755–760 110, ,Fntoa–tutrlpatmdlig e estl oli crop in tool versatile new A modelling: plant Functional–structural al., et 3–4 (1980). 131–148 31, .Ep Bot. Exp. J. etrso h hsooyo Plants of Physiology the on Lectures ,Ro rvtoimi euae yatasetltrlaxngradient auxin lateral transient a by regulated is gravitropism Root al., et h oe fMvmn nPlants in Movement of Power The M mie ’itieNtrle ehrhssrlUaedsFeuilles des l’Usage sur Recherches Naturelle. d’Histoire emoires ´ flnepyilge i aduhdrLhevmSofehesund Stoffwechsels vom Lehre der Handbuch Ein Pflanzenphysiologie: l,F ilaco icmuainadgairps as otwaving root cause gravitropism and Circumnutation Migliaccio, F. oll, ¨ 1121 (2009). 2101–2115 61, n.J e.Biol. Dev. J. Int. .Ep Bot. Exp. J. htte a erpoue rmtemodels the from reproduced be can they that .Ep Bot. Exp. J. LSCmu.Biol. Comput. PLoS W nemn,1904). Engelmann, (W. eini aue erigfo Trees from Learning Nature: in Design 1526 (2014). 2155–2165 66, 4–5 (1995). 143–150 46, 6–7 (2005). 665–674 49, .Ep Bot. Exp. J. rc al cd c.U.S.A. Sci. Acad. Natl. Proc. Apeo,1897). (Appleton, 1007(2015). e1004037 11, Ofr nvriyPes nls ed., English Press, University (Oxford 6–8 (2009). 461–486 60, rc al cd Sci. Acad. Natl. Proc. Srne Science (Springer nu e.Plant Rev. Annu. 4668–4673 109, SI 5 .Mrz .Prt .Tdn,M aldn,P act,Agnrl3 oe o growth for model 3d general gravitropism A Marcati, shoot P. Palladino, plant M. Tedone, of F. Porat, modeling A. Meroz, unifying Y. A 25. Moulia, B. Douady, S. Bastien, Investigations R. tropisms: Root 24. Aronne, G. Zanten, Van M. Izzo, G. L. plas- Muthert, F. DNA W. of L. Supercoiling 23. Neukirch, S. Heijden, der van M. H. G. Thompson, T. M. J. 22. 1 .Shik oeigspreia N:Rcn nltcladdnmcapproaches. dynamic and analytical Recent DNA: superhelical Modeling rods. Schlick, elastic T. twisted of 21. bacte- dynamics the of of modeling applications Biological the Klapper, and I. writhe 20. and twist of “Dynamics Klapper, I. Tabor, and M. growth Differential III: 19. rods Morphoelastic Goriely, A. Lessinnes, T. Moulton, E. D. 18. Goriely, A. enlightenment. 17. towards Bending Phototropism: Hangarter, P. R. Whippo, W. C. 16. 5 .d oca,H Louis, H. Monceau, du D. 15. iso ouehsflwrprmtrzto nFg n h eiwr for reviewers Grant the and research 6 Fig. under in Britain parameterization criticism. flower constructive J Great their his thank use We of to acknowledged. mission Council gratefully is Research EP/R020205/1 Sciences ical ACKNOWLEDGMENTS. (https://doi.org/10.5287/bodleian: Archive EoydP5kOP). Research University Oxford the in Availability. 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