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# 3D imaging and mechanical modeling of helical buckling in *Medicago truncatula* plant roots

Edited by William R. Schowalter, Princeton University, Princeton, NJ, and approved September 5, 2012 (received for review June 4, 2012)

## Abstract

We study the primary root growth of wild-type Medicago truncatula plants in heterogeneous environments using 3D time-lapse imaging. The growth medium is a transparent hydrogel consisting of a stiff lower layer and a compliant upper layer. We find that the roots deform into a helical shape just above the gel layer interface before penetrating into the lower layer. This geometry is interpreted as a combination of growth-induced mechanical buckling modulated by the growth medium and a simultaneous twisting near the root tip. We study the helical morphology as the modulus of the upper gel layer is varied and demonstrate that the size of the deformation varies with gel stiffness as expected by a mathematical model based on the theory of buckled rods. Moreover, we show that plant-to-plant variations can be accounted for by biomechanically plausible values of the model parameters.

Plant growth and crop productivity depend on the ability of plant root systems to secure water and nutrients from the heterogeneous terrestrial environment in which they grow. Soil compaction resulting from agricultural activities or from environmental changes such as drought impedes root growth and consequently has severe negative effects on yield (1). As world population continues to rise, plant breeding programs are challenged with the need to increase crop yields while facing a decline in agricultural soil quality including increased mechanical impedance of soil. Thus, there is a need to better understand the strategies that roots employ to grow in mechanically heterogeneous environments. Pioneering investigations have described the buckling of roots traversing air gaps in soils (2⇓–4) and measured the forces generated during root growth (5⇓⇓–8); however, further progress has been hindered by the opaque nature of soil.

Here, we build on recent imaging techniques (9⇓⇓⇓⇓–14) to investigate the growth of roots through mechanically heterogeneous environments. Our apparatus is distinct in that it employs a laser sheet and a translational stage to rapidly scan the region of root growth. Using this three-dimensional (3D) time-lapse imaging system, we observe primary *Medicago truncatula* roots growing through a transparent hydrogel composed of a compliant upper layer and stiff lower layer. The structural heterogeneity in the growth medium allows us to mechanically perturb the root in a controlled fashion. Consistently, we find the roots deform into a helical shape before penetrating into the lower layer as shown in Fig. 1. Because the length of the helical region is comparable to the length of the elongation zone in *Medicago* plants, it may be supposed that this morphology is purely a biological process such as circumnutation. However, our analysis reveals that (*i*) when the root encounters the stiff lower layer, tissue near the root tip twists via a remodeling process, and (*ii*) the mechanical buckling of the twisted root within the gel accounts for the observed helical shape. Collectively, these results demonstrate an important example of the interplay between mechanics and morphology during root growth in heterogeneous environments.

## Experimental Procedures

### Helical Root Growth.

A two-layer medium 8 cm thick was prepared using a transparent isotropic nutrient gel (15) solidified with two different concentrations of Gelrite. Using an Anton Paar rheometer, we measured the shear modulus and found *G*_{B} ≈ 1,500 Pa for the bottom layer and *G* ≈ 400 Pa for the top layer (see *SI Text*). Thus, the abrupt increase in stiffness at the gel/gel interface forms an elastic mechanical barrier to root growth. Through most of the top layer, roots grew straight down; if present, any root circumnutation was too subtle to observe. Just above the interface, however, we observed pronounced helical root deformations as shown in Fig. 1. Repeating the experiment, we found the general root morphology was reproducible, though each time there were variations in the shape and size of the helical deformations. From our visual inspections, we also noted that 74% of the root helices were right-handed, whereas the remaining 26% were left-handed (estimated uncertainty ± 9%).

### Mechanical Interpretation.

Based on our observations, we interpret the helical deformation as a form of mechanical buckling that occurs when the tip’s motion is halted by the stiff gel while the root continues to elongate. To investigate how this driving mechanism can lead to the observed root shapes, we developed a simple experimental model consisting of an axially compressed metal rod as a mechanical analog for the root. The rod, a nylon-coated stainless steel wire 0.4 mm in diameter and 8 cm long, was held vertically with the top end fixed to a stationary plate using epoxy. Axial force was manually applied with tweezers by pushing the lower end upward to compress the filament. When the wire is compressed in air, the resulting long wavelength deformation shown in Fig. 2*A* is consistent with the expected Euler buckling; the deformation occurs in a plane (Fig. 2*A*, *Inset*) and extends the full length of the wire.

In contrast, the root deformations were localized close to the tip (Fig. 1). To produce this effect in the mechanical model, we embedded the same wire in gelatin (Jell-O) and again applied axial compression from the lower end leading to reversible deformations. As shown in Fig. 2*B* and *Inset*, localized planar buckling was induced near the region where force was applied. This can be understood in the following manner. Buckling takes a sinusoidal form if the longitudinal stress is uniform throughout (16). However, the wire surface and gel adhere without slip so that displacements of the wire lead to shear deformations of the gel. Force balance shows that this gel shearing force accumulates along the wire’s length, reducing the wire’s internal longitudinal stress and attenuating the applied force. Thus, localized buckling occurs when a finite portion of the wire is above the buckling threshold while the rest is below (17). In order to produce a 3D helical shape, we added one additional feature: We manually twisted one end of the wire during compression. As Fig. 2*C* and *Inset* show, this combination of compression and twisting within a supporting medium produced a localized helical deformation like the shape observed in roots.

To check whether a similar twisting mechanism is at play during helical root growth, we fluorescently stained the epidermis of roots with a solution containing 10 μg/mL of 5-(4,6-dichlorotriazinyl)aminofluorescein and imaged them with a confocal microscope. For roots grown in unlayered gels, cell files were aligned vertically in columns along the entire length of the root (Fig. 3*A* and *Inset*). In layered gels where the roots encountered the stiff lower layer, cell files were twisted around the axis of the root in the helical region (Fig. 3*B*), and untwisted everywhere else. The localization of twisting shows that unobstructed root growth generally occurs without a visible preexisting chirality. Moreover, we extracted and compressed several straight roots, observing planar nonlocalized buckling in all cases, ruling out internal helicity as a twisting mechanism. Additionally, the distribution of handedness for the root morphology shown in Fig. 1*F* demonstrates that passive physical instabilities such as those seen during the coiling of poured viscous liquids (18) are insufficient for generating root twisting because they would lead to, on average, equal numbers of either handedness. Finally, we note that differential elongation as currently understood would only produce planar buckling.

Collectively, these observations suggest that the twisting involved in helical buckling arises from a touch-activated biological remodeling in response to axial loads. Though the microscopic dynamics were not observable in our experiments, the process may be related to the thigmotropic-modulated gravitropism previously reported in *Arabidopsis* (19).

## Model

The wire model provides a qualitative understanding of helical root buckling. It is unclear, however, whether the mechanistic interpretation of the root as a twisted buckled rod embedded in a gel can capture the plant-to-plant variations typically seen in our experiments (Fig. 1). Toward this end, we (*i*) developed an experimental protocol for measuring variations in root morphology, (*ii*) developed a mathematical formulation of the buckled rod model, (*iii*) fitted the model to the data, and (*iv*) determined whether the fitted values of the model parameters are biomechanically plausible.

### Measurements of Root Geometry.

To quantify the plant-to-plant variations in root morphology, we measured the shape of the helical regions using a unique imaging technique we developed for 3D time-lapse imaging of growing roots (3D-TIGR). In essence, our apparatus (Fig. 4) scans the region of root growth with a laser sheet while taking image slices spaced every 0.150 mm. Each scan took less than 5 min. The image slices were then processed in IMARIS 6.0 to create a 3D reconstruction of the root and to extract its spatial coordinates .

The total imaging time for each root growth experiment was approximately 100 h. To establish an experimental protocol for measurement of the helical morphology, we recorded time-lapse movies for 13 roots at a rate of one 3D scan per hour (Movies S1–S3). These movies reveal that when steady growth is impeded by the stiff gel, the root abruptly deforms in the transverse direction as expected for a buckled rod. Moreover, because the shape is already helical, root twisting must initiate before buckling. Continued growth leads to the stereotypical shapes shown in Fig. 1. After the tip penetrates into the lower layer, the radial extent of the helix rapidly shrinks by 30 ± 10% (Fig. S1). Thus, for the following analysis, we scanned each root after it passed through the barrier and then linearly scaled the transverse size of the helix by 1.43 to recover the buckled shape before penetration.

To quantify the size of the helical deformation, we defined two length scales, the average vertical extent of the helix , and the average squared radius of the helix . These longitudinal and transverse measures are depicted schematically in the *insets* of Fig. 5. We calculated and from the scaled 3D-TIGR root coordinate data using [1]where *r*^{2} = *x*(*z*)^{2} + *y*(*z*)^{2} is measured from the central axis of the helix, which was oriented to coincide with the *z* axis. Bounds of the integrals were defined by noting the curvature of *r*(*z*) is zero outside the helical region; however, we note the equations in Eq. **1** are generally insensitive to the choice of endpoints.

For the gel system shown in Fig. 1 where *G* ≈ 400 Pa, there were substantial root-to-root variations in both and (Fig. 5). To gain further insight into the range of possible root morphologies and their dependence on *G*, we grew 67 plants in gels where the top layer modulus was varied from about 100 to 1,500 Pa. For these gels, we saw no apparent dependence of the root radius or length on the modulus; however, and were found to depend inversely on *G* (Fig. 5). Furthermore, the spreads in and at fixed *G* are also inversely related to the modulus: At *G* ≈ 250 Pa there is a three-fold variation in and over an order of magnitude variation in , whereas at *G* ≈ 1,500 Pa these variations are significantly reduced. Evidently, the root geometry and its variations are strongly dependent on the stiffness of the growth medium.

### Theoretical Model: Development and Quantitative Fitting.

For simplicity, we modeled the root tissue from the helical region as a homogeneous inextensible isotropic cylindrical rod. These assumptions are consistent with experimental observations: (*i*) Root cells are roughly 10^{2} times smaller than the typical dimensions of the helix (Fig. 3), (*ii*) the time scale for growth is much longer than the buckling instability time scale (Movies S1–S3), and (*iii*) neither the material properties nor root radius vary significantly over the length of the helical region (20). These assumptions allow the rod to be described by a constant bending modulus *EI* with Young’s modulus *E* and moment of area *I*. Next, we embed the theoretical rod in a linear elastic gel with shear modulus *G*. Based on the growth of fine hairs that anchor the root to its growth medium (8, 20), we assume a no-slip boundary condition. For simplicity, we neglect viscoplastic effects in the gel; rheological measurements, elastic relaxation of roots after penetration into the lower gel, and the absence of cavitation bubbles support this assumption. Finally, we specifically focus on variations in root morphology and therefore exclude the dynamic components of touch-activated twisting from our model.

The general data trends can be understood by basic scaling arguments. A force *T*_{0} greater than the critical buckling force *F*_{c} causes a rod to buckle into an arc of length *L* with amplitude *u* and bending energy ∼*EI*(*u*/*L*^{2})^{2} × *L*. The buckled rod causes a volume ∼*L*^{3} of the embedding gel to deform with an energy ∼*G*(*u*/*L*)^{2} × *L*^{3}. For a fixed force *T*_{0}, we minimize the sum of these energies with respect to arc length to find *L* ∼ *ℓ*, where the characteristic length scale *ℓ* ≡ (*EI*/*G*)^{1/4}. Therefore, we expect . Furthermore, the scaled transverse displacement increases with the scaled excess force (*T*_{0} - *F*_{c})/*F*_{0}, where the characteristic force scale *F*_{0} ≡ [(*EI*)*G*]^{1/2}. Thus, will have an inverse dependence on *G* due to the factor of *ℓ*^{2} and because *F*_{0} is larger in stiffer gels. These arguments for and predict smaller root deformations in stiffer gels. Although the experimental measurements qualitatively agree, the data have significant scatter and are too limited in range to test the predicted scaling.

Using a more detailed application of the theoretical rod model, we test whether mechanical buckling can account for the entire morphology of each root as well as the individual variability. Parameterizing the centerline of the rod as , the key mechanical quantities of interest are the longitudinal compressive force *T*(*z*) and the axial moment *M*_{z}(*z*). Within the small deflection approximation where the infinitesimal element of arclength *ds* ≅ *dz*, the equations of equilibrium for the transverse forces per unit length are [2]Primes indicate differentiation with respect to *z*, and α ≈ 2*G* is the effective transverse spring constant per unit length due to the gel elasticity (see *SI Text* for detailed derivation). In each equation, the left-hand side includes terms for (*i*) the bending force of the rod, (*ii*) the force required for torque balance when the centerline is twisting, and (*iii*) the projection of *T*(*z*) along the rod’s path. To calculate the dependence of and on the gel modulus, we first determine *T*(*z*), *M*_{z}(*z*), and the appropriate boundary conditions.

We find the compressive force *T*(*z*) by considering growth just prior to buckling when the tip has made contact with the stiff lower gel. Because growth is obstructed, root elongation, which occurs at the tip, leads to a uniform longitudinal upward displacement of the entire root. However, fine hairs anchor the root to the embedding gel, leading to a downward linear restoring force acting on each portion of the root. Assuming no slip between the root and gel, force balance yields [3]where *T*_{0} is the force applied on the root tip by the lower gel and *Z* is the length of the root. Eq. **3** models the nonuniform compressive force previously discussed.

In roots, the moment *M*_{z}(*z*) arises from the response of individual cells to their local loading conditions and likely results in a remodeling of the root’s elastically unstrained reference state. Because the in vivo details are unknown, we take a phenomenological approach and calculate the required moment to produce a given helical morphology by integrating the equations of equilibrium over 20 experimentally measured root contours (see *SI Text*). Generally, we find the moment is zero outside the helical region, and nonzero within (Fig. S2). Following this trend, we approximate the functional form as [4]*M*_{0} has two contributions: (*i*) the previously discussed remodeling of root tissue, and (*ii*) the root’s intrinsic elasticity. Although the latter contribution gives rise to a twist per unit length Δτ beyond the remodeled reference state, both are related to the observed cell file twisting. If elasticity dominates the moment, then *M*_{0} = *C*(*z*)Δτ(*z*), where for a homogeneous isotropic inextensible rod, the torsional rigidity *C*(*z*) = (2/3)*EI* = constant (21). Thus, the expression Δτ(*z*) = 3*M*_{0}/2*EI* = constant is a bound on the rate of cell file twisting.

To determine the boundary conditions and test the model, we performed an iterative nonlinear least-squares fitting of the root coordinate data from the helical region to Eqs. **2**–**4**. Specifically, we use the Levenberg–Marquardt algorithm to determine the best-fit values for the model parameters *EI*, *T*_{0}, and *M*_{0}, as well as the transverse forces and moments at both ends of the fitting interval. This process was repeated for all roots; 49 of the 67 plants had convergent fits (Fig. 6*A*). Of the nonconvergent fits, an inspection of the complete 3D-TIGR data revealed helical morphologies that violated the small deflection approximation.

From the convergent fits, we infer the appropriate boundary conditions. Generally, the transverse forces *F*_{x} ∝ *x* and *F*_{y} ∝ *y* vanished at both ends. Similarly, the transverse moments *M*_{x} ∝ *y*^{′′}, and *M*_{y} ∝ *x*^{′′} were smallest at *z* = 0, while the tangent components *x*^{′} and *y*^{′} vanished at *z* = *Z*. Collectively, these results yield a hinged boundary condition (*x*,*y*,*x*^{′′},*y*^{′′} = 0) at the bottom of the rod, and a clamped boundary condition (*x*,*y*,*x*^{′},*y*^{′} = 0) at the top. Additionally, we found the majority of best-fit values for *EI* and *T*_{0} were spread over two orders of magnitude (Fig. 6 *B* and *D*, yellow crosses), while the estimated twist per unit length 3*M*_{0}/2*EI* was clustered between 0.1 and 1.0 rad/mm (Fig. 6*E*, yellow crosses).

### Independent Checks of Fitted Parameters.

To check whether the values for the fitting parameters are biomechanically plausible, we independently estimated *EI*, *T*_{0}, and *M*_{0}. Starting with the bending modulus, we measured *EI* for 16 roots in a three-point bending apparatus (see *SI Text*). Unfortunately, root tissue from the helical region was too short and fragile to work with. Therefore, we made measurements on the older, more lignified root tissue between the helical growth and the base of the plant. Because this tissue was typically 5–7 days old, it was thicker and easier to work with. Indeed, we measured *EI*_{M} = (3.5 ± 1.6) × 10^{-7} Nm^{2} (SD) (Fig. 6*B*, black solid and dashed lines), which agrees with the upper range predicted by fitting.

Concerning root-to-root variations, it is unlikely that Young’s modulus *E* varies enough to account for the spread in the fitted *EI*. However, if each root has a distinct radius ρ in the helical region, there can be significant variation in the moment of area *I* = (π/4)ρ^{4} (Fig. 6*B*, green contours). To investigate this possibility, we assumed *E* was constant and that differences between the fitted *EI* and measured *EI*_{M} were due solely to ρ. We then calculated the predicted reduction in root radius relative to mature tissue, (*EI*/*EI*_{M})^{1/4}. From the raw 3D-TIGR data, we measured the average root radius in the helically buckled and basal regions to find . Comparing these quantities, we find a strong correlation confirming that variations in the root radius can account for spread in the fitted *EI* (Fig. 6*C*).

Although *T*_{0} is difficult to measure experimentally, estimates of its value can be made from deformations in the gel interface induced by the root tip. Detailed calculations (21) show that a point force *T*_{D} on a half-infinite elastic medium causes a dimple of depth *D* and radius ρ. Here, ρ is the same as the root tip radius. Because we have two elastic mediums, *T*_{D} = 4π(*G* + *G*_{B})*D*ρ, where *G* and *G*_{B} are the shear moduli of the top and bottom gel layers, respectively. Visual observations show *D* ≈ 2 ± 1 mm, and ρ ≈ 0.50 ± 0.25 mm.

Assuming *T*_{0} ≅ *T*_{D}, we estimate the tip force along with upper and lower bounds as a function of the top gel modulus (Fig. 6*D*, black solid and dashed lines). Values range from 5 to 100 mN and agree with 80% of the fits, consistent with the possibility that some of the scatter in *T*_{0} arises from variations in *G*. Additional estimates based on a Hertz contact or the gel fracture strength are consistent with these results (see *SI Text*). Deviations from theoretical expectations can be accounted for by imperfect coupling between the root and the gel or variations in the root tip’s angle of attack resulting in a decreased normal force on the gel surface.

To check the range of best-fit values for the moment *M*_{0}, we used confocal images to measure the cell file angle with respect to the root axis in the helically buckled region. Imaging several roots, we found an average twist of τ_{M} = (2.1 ± 0.7) radians/mm (SD) (Fig. 6*E*, black solid and dashed lines). Comparing with the fits, we see τ_{M} overestimates Δτ = 3*M*_{0}/2*EI*. This overestimate can be attributed to remodeling of the unstrained reference state wherein an elastically relaxed configuration still exhibits twisted cell files. Root-to-root variation in *M*_{0} can be attributed to differences in the growth rate of individual plants.

Collectively, the range of best-fit values for the bending modulus *EI*, the tip compressive force *T*_{0}, and the moment *M*_{0} are consistent with our independent checks and thus biomechanically plausible. These findings demonstrate our simplified mathematical model is capable of quantitatively accounting for the variations observed in the root morphology.

### Relating Model Parameters to Root Morphology.

To identify the connection between variation in specific model parameters and root morphology, we used Eqs. **2**–**4** to simulate the dependence of and on *EI*, *T*_{0}, and *M*_{0}. Specifically, we performed sets of numerical solutions while systematically varying the model parameters within the ranges determined by fitting. In our simulation, we increased *T*_{0} until the rod buckled, at which point we evaluated and from the solution .

Over the experimental range of *G*, we found that depends primarily on *EI*, depends primarily on *T*_{0}, while neither depends strongly on *M*_{0}. Specifically, we fixed *T*_{0} = 10 mN, 3*M*_{0}/2*EI* = 0.45 radians/mm, and varied *EI* over the range illustrated by the green contours in Fig. 6*B*, producing a corresponding set of contours for as a function of *G* (Fig. 5*A*). Similarly, we fixed *EI* = 2.2 × 10^{-8} Nm^{2}, 3*M*_{0}/2*EI* = 0.45 radians/mm, and varied *T*_{0} over the range illustrated by the green contours in Fig. 6*D*. This produced a set of contours for as a function of *G* (Fig. 5*B*). The dependence of on *EI* and on *T*_{0} was negligible and could not account for variations at fixed *G*. Finally, we fixed *EI* = 2.2 × 10^{-8} Nm^{2}, *T*_{0} = 10 mN, and varied 3*M*_{0}/2*EI* from 0.1 to u0.7 radians/mm to produce a set of contours (Fig. S3) that showed weak sensitivity of and on *M*_{0}.

From the contours in Fig. 5, we are able to read off the scaling relations for and . We find , and , where *ℓ* and *F*_{0} are the length and force scales, respectively, defined previously in the scaling arguments. Indeed, these numerically determined expressions agree well with theoretical expectations.

## Conclusions

Using 3D-TIGR, we studied the helical buckling of *Medicago truncatula* roots due to a physical barrier in their growth medium. This morphology could impact the fitness of *Medicago* plants in at least two ways. First, the helical geometry converts axial loads into transverse loads, allowing the root to brace against the surrounding medium and generate a greater force at the tip. Second, touch-activated twisting induced by impenetrable barriers such as rocks leads to a mechanical instability that redirects root growth along the surface of the obstruction (22). Thus, helical buckling could enhance the root’s ability to force through or around physical barriers, allowing greater access to resources in its environment.

Finally, we speculate that the root geometry observed here may be related to the skewed sinusoidal growth pattern known as root waving, in which roots growing on tilted 2D surfaces oscillate rather than growing straight down the slope (23⇓–25). Though further experiments are necessary, we may discover in time that this growth behavior, along with other plant morphologies, have explanations rooted in the mechanics of growing materials.

## Acknowledgments

The authors thank A. Moore, L. Ristroph, J. Savage, Z. Chen, L. Manning, M. Lapa, M. Haataja, J. Sethna, A. Alemi, B. Davidovitch, C. Orellana, E. Kolb, the Cohen lab, and the Mahadevan lab for stimulating conversation. We also thank J. Gregoire and S. Iams for assisting in apparatus development, J. Fetcho for kindly allowing us to use Imaris, M. Venkadesan for kindly loaning the translation stage, and J. Puzey for critically reading this manuscript. This work was supported by the National Science Foundation through a Graduate Research Fellowship to J.L.S., Grant IOS-0842720 supporting R.D.N., Grant DMR-1056662 supporting I.C., and Cornell’s IGERT Program in Nonlinear Systems (National Science Foundation Grant DGE-9870631) supporting S.J.G.. This work was supported by the US Department of Energy through Grant DE-FG02-89ER-45405 supporting M.S.P. and C.L.H., and Grant DE-FG02-08ER46517 supporting S.J.G.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: JLS533{at}cornell.edu. ↵

^{2}Present address: Department of Plant Pathology, North Carolina State University, 2510 Thomas Hall, Raleigh, NC 27695-7616.↵

^{3}Present address: Department of Physics, Harvey Mudd College, 301 Platt Blvd., Claremont, CA 91711-5990.

Author contributions: J.L.S., M.J.H., C.L.H., I.C., and S.J.G. designed research; J.L.S., R.D.N., M.S.P., and S.J.G. performed research; J.L.S., C.L.H., I.C., and S.J.G. analyzed data; and J.L.S., M.J.H., C.L.H., I.C., and S.J.G. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1209287109/-/DCSupplemental.

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