A Unified Model of Shoot Tropism in Plants: Photo-, Gravi- and Propio-ception
[['Plants are constantly moving to reach the light and to maintain the architecture and posture of their aerial organs: stems , branches , leaves… These movements are active , generally powered by differential growth and are controlled by environmental signals [1] .', 'Tropisms are the class of movements that are oriented by a vectorial environmental factor .', 'Light is the main source of energy for plants and is a major cue for tropic movement .', 'In phototropism , shoots grow in the direction of the light source .', 'Under natural conditions on Earth , gravity is unavoidable and drives gravitropism , in which shoots usually grow against the direction of gravity .', 'Proprioception , the ability of plants to perceive their own deformations , has recently been identified as a major factor in tropic movement; it stimulates the straightening of a curved organ , a response that can be thought of as autotropism [2–4] .', 'Such classification of control mechanisms into three different tropic drivers is however merely a conceptual convention: it is likely that , in natural conditions , the three processes interact constantly .', 'Plant shoots generally exhibit negative gravitropism , positive phototropism , and negative autotropism in the same organs at the same time .', 'The way phototropism and gravitropism interact to control the movement of plant shoots is acknowledged to be important in determining plant habit in natural conditions but is not well understood [5] .', 'Advanced genetic studies in the model species Arabidopsis thaliana , which used mutants with impaired gravitropism or phototropism [6 , 7] , have shown that the two processes interact in a complex manner to control movement in wild-type plants [8] .', 'However , part of this complexity lies in the variety of phototropic responses observed in response to different qualities and intensities of the light stimulus .', 'For example , the effects of low-fluence pulses differ from those of continuous light [7]; but only the latter are directly relevant to plant growth under natural conditions , and will therefore be considered here .', 'The focus of the current study is the regulation and the control of organ movements during the interaction of tropisms .', 'To do so , it is proposed to extend the recent dynamical model for gravitropism called the AC model [2] to phototropism .', 'In this model , active tropic bending is controlled by the additive ( but opposing ) effects of graviception and proprioception , expressed by', '∂C ( s , t ) ∂ t = - β A ( s , t ) - γ C ( s , t ) for s > L - L g z and 0 otherwise ( 1 ) where s is the position along the organ , Lgz is the length of the growth zone , L is the length of the entire organ , t is time , A ( s ,', 't ) is the local angle of the organ to the vertical , C ( s ,', 't ) is the local curvature ( i . e . , the spatial rate of change of A along', 's ) and the parameters β and γ are , respectively , the gravi- and proprio-ceptive sensitivities .', 'The AC model was shown to explain the complex kinematics of gravitropism in eleven species covering a broad taxonomical range of angiosperms , major growth habits and organ types .', '[3 , 4] Although graviception and proprioception act additively , their control over the dynamics of tropic movement and the steady-state final shape actually depends only on the ratio between gravisentitivity and propriosentitivity , scaled to the size of the growth zone .', 'In the model , this is formalized through the definition of the dimensionless graviproprioceptive bending number B = β Lgz/γ .', 'B fully defines both the time to reach the steady state and the final shape at steady state , and can be measured in simple morphometric experiments .', 'As B is dimensionless , it can be used to make quantitative comparisons between experiments involving very different sizes and growth velocities .', 'This enables universal behaviors and control mechanisms to be identified .', 'To understand and describe properly the interaction between gravitropism and phototropism , successive steps are undertaken: First , the hypotheses that lead the construction of the model are discussed , then general specifications of the geometry of the organ and of the gravity and light fields are established .', 'In the following part , the construction of model is given .', 'Starting with the simple case of phototropism in isolation which allows to dissect the interaction between photosensitivity and propriosensitivity .', 'Finally , the dynamics of the interaction among photoception , graviception and proprioception are explored , including the influence of light intensity .', 'All these models are investigated for different localization of photoception , at the tip or distributed along the organ .', 'Quantitative experimental data on photogravitropic equilibrium are used to test the validity of the model .', 'These results and the implications for plant biology are discussed .'], ['In order to formulate hypotheses regarding the extension of the AC model to include phototropism , it is now useful to review our knowledge of the distribution of sensing mechanisms and differential growth responses along the plant organ , the mechanisms of photo- gravi- and proprio-ception , as well as their possible interactions .', 'The localization of gravisensitivity has been established in more detail than the localization of photosensitivity .', 'The plant perception of gravity is related to the presence of statoliths within specialized cells called statocysts .', 'In aerial organs , statocysts and statoliths are found throughout the growth zone , and both perception and the bending response are local [9] .', 'Our knowledge of the mechanisms of proprioception is limited , but it is known to be exclusively local and likely to involve cytoskeleton remodeling [1 , 2] .', 'In phototropism , blue-light-sensitive phototropins sense the direction of incoming light , but they interact with other photoreceptors in a network that remains to be fully elucidated [10–13] .', 'Photoreceptor localization studies were pioneered by Darwin [14] .', 'By masking different parts of coleoptiles , Darwin found that the perception of light occurred at the coleoptile apex .', 'As tropic bending occurred all along the growth zone of the organ , a secondary basipetal signal was postulated to be involved , now elucidated as a lateral redistribution of the polar transport of the plant hormone auxin [10] .', 'Very few studies have further systematically analyzed the localization of photoreceptors or even phototropic responses along plant organs .', 'It has been found , however , that apical phototropic sensing is not universal .', 'For example , the hypocotyl of Arabidopsis exhibits photoreceptors all along the growth zone , allowing for distributed local photoception and local phototropic growth responses [11 , 15] .', 'The functional significance of such differences in the localization of photosensitivity among organs has not been investigated .', 'A few quantitative studies of the effects of continuous light [16–18] have shown that phototropism and gravitropism seem to act in an additive manner .', 'When a plant organ is lit from a direction that differs from that of gravity , the plant undergoes an active movement until a steady-state shape called the photogravitropic equilibrium is reached [17 , 19] .', 'The photogravitropic equilibrium angle ( PGEA ) of the organ tip was found to follow the phenomenological equation', 'P G E A = k log I I 0 - g sin A 0 ( 2 ) where I is the fluence rate of illumination , I0 is the light-sensing threshold , g is the gravitational force , and A0 is the initial angle of inclination towards gravity .', 'The term gsinA0 is the sine law of gravitropism [4] .', 'This sine law was also used when defining the gravisensing term in the AC model for gravitropism1 but in this case it has been reduced to a linear term using the approximation sin ( A ( s , t ) ) ≈ A ( s , t ) + O ( A3 ) .', 'The term k log ( I/I0 ) reflects the phototropic stimuli .', 'Both phototropic and gravitropic sensing have been shown to control the relocalization of polar auxin transporters , controlling the formation of lateral gradients in auxin concentration and hence differential growth .', 'These molecular dynamics might explain why the effects of the two stimuli on PGEA are additive [7 , 20 , 21] .', 'Thus far , the dynamics of the processes leading to this photogravitropic equilibrium have not been analyzed or modeled in detail .', 'Building on this current knowledge , we propose herein to extend the graviproprioceptive AC modeling approach to study phototropism and its interactions with gravitropism .', 'We focus on investigating dynamic control of photosensitivity , gravisensitivity and propriosensitivity throughout the whole movement .', 'More precisely , our working hypotheses are as follows: ( H1 ) The action of the tropic motor is fully driven by the perception-regulation process and results in a change in the local curvature .', '( H2 )', 'The angles formed between the axis of the growing organ and the respective gravity and light fields influence graviception and photoception , and hence influence gravity- and light components driving the tropic response .', '( H3 )', 'Proprioception takes place regardless of whether external tropic signals are present .', 'The occurrence of proprioception has been shown for gravitropism [2 , 22] .', 'Concerning phototropism , plants growing in a clinostat ( an apparatus that suppresses graviception ) were also shown to straighten after a transient light pulse [23] .', 'Thus , we assume that each constituent element of the organ perceives its own local deformation during active bending , namely the curvature , and responds in order to restore local straightness [2] .', '( H4 )', 'The different types of perception act additively within the model , meaning that their respective contributions have equivalent roles in driving the movement .', 'It is therefore expected that much of the apparent complexity of the motion is due to spatio-temporal integration of several responses over the changing geometry of the organ with respect to the light and gravity fields .', 'In order to formulate testable and refutable predictions regarding the dynamic control of tropic movement , we combine these working hypotheses into successive variants of a dynamic model , extending the graviproprioceptive AC modeling approach .', 'These models are expected to provide relevant dimensionless numbers that control the dynamics , as well as a means of estimating their values on the basis of experimental measurements .', 'This will enable the model and hypotheses to be evaluated experimentally , and eventually open the way to obtaining accurate phenotypes of tropisms .', 'We have chosen to neglect the effects related to the elongation of the organ .', 'Recent work has shown that elongation tends to destabilize tropic movements and should increase the oscillations that take place during these movements [24] .', 'We have found , however , that the proprioceptive sensitivities of modern angiosperms have evolved such that these effects are fully controlled by the plants .', 'Therefore , these growth-related effects can be neglected in the description of tropic movement [24] .', 'Similarly , even though plants are expected to bend under their own weight , we propose that the effects of a plant’s weight on its shape can also be neglected [25] .', 'This is based on the fact that the AC model , which takes into account the plant’s perception as a sole driving process , has proved to be sufficient to explain the movement of a wide variety of organs and species of many orders of magnitude of size .', 'To finish , it is important to note that the AC model accounts for orthogravitropism , wherein the organ aligns with the direction of gravity .', 'Some organs align in a direction different from that of gravity , called the gravitropic set-point angle ( GSA ) [26 , 27] .', 'It is possible to adapt the AC model for such cases by modifying the model’s graviceptive term , −β A ( s , t ) →−β ( A ( s , t ) −GSA ) such that the organ aligns with the GSA .', 'However , the AC model has not been tested for non-orthogravitropic organs .', 'And this study concentrates on the case of ortho-gravi and photo-tropisms ( a very common case in primary shoots ) .', 'As stated before , the present study involve successive variants of the model .', 'To unease their handling we named the successive models presented below according to', 'i ) the sensory processes that are involved and their locations ( biological specification ) —e . g . , photoproprioceptive; and', 'ii ) the driving variables under investigation ( mathematical specification ) ( in accordance with the naming convention in [2] , e . g . model AC is driven par the inclination angle A versus the vertical and the curvature C all along the axis , whereas model AaC is driven by the apical angle of inclination and by the curvature all along the axis ) .', 'The reader may find a list of name of the variables and parameters in Table 1 and of the model in Table 2 .', 'One drawback of most studies on phototropism is that they have overlooked the true geometry of the organ , measuring only the orientation of the apical tip of the organ [28–31] .', 'One of the greatest insight of the AC model was to show that the geometry of the organ and of the source field is central to describe the movement properly .', 'It is then important to have a clear description of the studied geometry .', 'Like the AC model , the models developed herein describe the shape of the organ in terms of its median line , i . e . , its central axis ( Fig . 1 ) .', 'We parameterize the position along this median line according to the curvilinear abscissa s going from the base s = 0 to the apex s = L . The angle A ( s , t ) then describes the local orientation of the median with respect to the vertical pointing up ( and hence to the direction of the gravity vector g ) at time t .', 'The angle of the apex at the tip of the organ , Aa , is then equal to A ( L , t ) .', 'The local curvature C ( s , t ) is the spatial rate of change of A ( s , t ) along s , and from differential geometry we know that', 'C ( s , t ) = ∂ A ( s , t ) ∂ s or A ( s , t ) = A 0 + ∫ 0 s C ( l , t ) d l ( 3 ) At the scale of the plant , gravity acceleration g is homogeneous , uniform , and invariant to translation and to rotation around the main direction of the gravity field ( Fig . 1 ) .', 'The case of light perception seems to be less straightforward .', 'The geometry of light fields in nature can be quite diverse .', 'At the scale of the plant , the light field produced by the sun on the earth is supposed to be homogeneous , uniform , and invariant to translation and to rotation around the main direction of the light field , but its orientation relative to the gravity field varies over the course of the day .', 'In the lab , a point light source can emit a spherical field whose intensity ( irradiance ) decreases with the square of the distance .', 'Furthermore , multiple point sources might be used , thereby increasing the complexity of the analysis .', 'Thus far , most controlled experiments have used a distant punctual source with a collimated beam [16–18 , 32] .', 'For simplicity , we assume that the light field l has the same symmetry as the gravity field .', 'This assumption holds true if the source of light is far from the organ .', 'To further reduce the complexity of the analysis , we do not consider curvature outside the plane; that is , we explore the case in which the main direction of the plant’s organ , the gravity field g and the light field l are all in the same plane Pgl ( Fig . 1 ) .', 'This again was the case in most controlled experimental studies ( e . g . [17] ) .', 'Therefore , the direction of the lighting in the Pgl plane is defined by the angle Ap and by a radiance ( or light intensity ) I0 ( Fig . 1 ) .', 'The simplifications we adopt enable us to assess the model by comparing it with experimental data , while still providing insights regarding the regulation of the movement .', 'Moreover , the assumption that the initial direction of the organ , the direction of gravity and that of the light are in the same plane is fulfilled for organs that are already aligned with the direction of gravity .', 'Coleoptiles and other plant shoots appear to sense light direction via the light gradient across the organ [33] .', 'Treatments that change the steepness of this gradient ( e . g . , infiltration with dyes or changes in the amounts of natural pigments ) alter the phototropic response: the steeper the gradient , the more the coleoptile bends towards the light .', 'This gradient depends on internal light transfer through plant tissues but also depends linearly on the irradiance of the light impinging on one side of the surface of an organ .', 'This irradiance is given by Lambert’s cosine law of irradiance of geometrical optics ( e . g . [17] ) , which states that the irradiance falling on any surface varies in proportion to the cosine of the incident angle .', 'I ( s , A P ) = I 0 cos π 2 - ( A ( s , t ) - A P ) = - I 0 sin A ( s , t ) - A P ( 4 ) As the AC model is a first-order model , and in the limit of small angles A ( s , t ) −AP , we can use the approximation sin ( A ( s , t ) −AP ) ≈ A ( s , t ) −AP + O ( A3 ) , so that the photosensitivity is proportional to the angle between the organ and the light direction i . e . , A ( s , t ) −AP ( the influence of the intensity of the light irradiance will be considered in a later section ) .', 'Although this approximation is only valid for small angles , such an approximation gives a good and efficient understanding of the dynamics of the system .', 'The influence of each term on the tropic movement can be easily understood and discussed .', 'We may now start with the simplest case of phototropism without gravitropism .', 'Using equation 4 , a phototropic model can be constructed .', 'Two sub-models are considered , when the perception is apical and when the perception is local .', 'The previous model can now be easily extended to take into account the perceptions of both light and gravity .', 'As described in Fig . 1 . A , the direction of the light stimulus is now considered to form an angle AP with the vertical , the direction of the gravity .', 'The orientation angle A ( s , t ) along the organ is still measured with respect to the vertical , and hence to the direction of gravity .', 'If AP ≠ 0 the symmetry of the system is broken .', 'When only one of the two tropisms influences the system , rotation around the axis defined by the direction of the corresponding field should have no effect on the system .', 'When the two tropisms act simultaneously , however , the angle between the two fields prevents such global symmetry .', 'It should also be noted that if the gravitropic term or the phototropic term tends to 0 , the system converges to the photo-proprioceptive models outlined above .', 'As for the phototropic models , effects due to local and apical perception are considered and discussed .', 'A series of experiments were conducted or reprocessed from the bibliography in order to assess the Photo-Gravitropic models against experimental data in continuous light and 1g conditions ( i . e . reminiscent of natural outdoor conditions on Earth ) : an experiment in the typical set-up used for phototropic studies , but tracking the entire kinematics of the movement ( LLE experiment ) and a set of experiments conducted by [17] and focusing only on the PhotoGravitropic Equilibrium tip Angle ( PGEA ) , but at various tilting angles and light intensities ( referred to as “Galland’s experiments” in the following ) .'], ['A major prediction of the of AR-based models is that even when photoception and graviception act in different directions ( i . e . , AP ≠ 0 ) , they seem to act together to bring the organ towards a single direction defined by the angle AR .', 'Additionally the ARC model predicts that the whole coleoptile curves in the direction AR which is first achieved at the tip and the curvature then concentrates near the base . These two predictions are independent of the position at which light is perceived , i . e . , at the tip of the organ or along the entire organ , and is thus common to all the variants of the ARC model , ( i . e . the local photoception model ARC and apical photoception model A R a C and their versions including explicit light intensity effects AR ( I ) C and A R a ( I ) C ) .', 'For the sake of simplicity , whenever this common core is addressed , the notation ARC will be used to refer to all of them .', 'When illuminated by a source of light from a direction that differed from that of gravity ( LLE experiment ) , the coleoptile curved and finally reached a steady state .', 'The apical part aligned with an angle that was an intermediary between the direction of gravity ( A = 0 ) and the direction of the light ( A = π/2 ) ( Fig . 3 . A and B ) .', 'This shows qualitative agreement with the ARC model .', 'Additionnally the transient kinematic pattern in which', 'i ) the whole coleoptile curves until the direction AR is first achieved at the tip and', 'ii ) the curvature then concentrates near the base .', 'is qualitatively observed in ( Fig . 3 . B and C ) .', 'However , this kinematic experiment also revealed some limits of grass coleoptile as a “model system” for tropic studies .', 'Indeed , when a coleoptile is illuminated , photomorphogenetic effects take place , slowing down the expansion of the coleoptile and increasing that of the inner leaf .', 'After approximately 10h , the leaf inside the coleoptile pierces the coleoptile .', 'The dynamics of the coleoptile then ceases , and there is no further elongation of coleoptile .', 'The dynamics of the movement is then fully determined by the leaf that was inside and this dynamics differs from that of the coleoptile .', 'Moreover the influence of the inner leaf may explain why the rather steady tip angle ( between 3 and 6 h in the example Fig . 3 . A and B ) is subsequently varied again ( between 7 and 10h ) , and it is thus difficult to determine whether the coleoptile has had time to reach a steady state before been disrupted .', 'However , the apical tip is predicted in the model to converge to the direction defined by AR before the shape reaches a steady state , and we saw that this is consistent with what is observed experimentally .', 'We may thus assume that in Galland [17] estimate of AR could be measured even when the full steady state of the rest of the coleoptile could not be well defined .', 'We therefore tested further the validity of the functional form of AR-based models through the analysis of the apical tip angle at PhotoGravitropic Equilibrium ( PGEA ) in the work by Galland [17] .', 'If the ARC core-model is correct , then the experimental PGEA should be equal to AR , and according to equation 14 , it should depend on the angle between the direction of gravity and direction of the light AP , and on the ratio between the gravisensitivity and the photosensitivity levels .', 'Indeed AR can be expressed in the model as a function the photograviceptive number M:', 'A R = A P 1 1 + M ( 25 )', 'It follows directly that when photoception dominates , the organ bends in the direction of the light , M < < 1 , AR = AP .', 'However when graviception dominates , M > > 1 , AR = 0 ( Fig . 4 ) .', 'It is then possible to express M directly as a function of AR/AP .', 'M = A P A R - 1 ( 26 ) A direct quantitative assessment of the functionnal form of the AR ( I ) C and A R a ( I ) C models—and of the competing M ∼ Φ ( I ) sub-models—vs .', 'experimental data can then be conducted through the analysis of the experimental relation between the steady-state tip angle PGEA and the fluence rate of the incident light I . Indeed AP is known , and we have seen that we may assume AR = PGEA .', 'Then using equation 27 an estimate of M can be obtained .', 'And combining equation 27 with equation 23 , a prediction of the model is then that a plot of M = AP/AR − 1 as a function of I should display either a power-law dependancy , equation 24 , or a log-dependancy , equation 25 .', 'Moreover , these relations should fit to a single curve M ∼ Φ ( I ) , independent of the initial tilting angle A0 .', 'Note however that due to the linearization of the sine terms of the angle dependency of gravi- and photoception ( equations 1 and 4 ) this should be only valid for inclination angles A0 < 90° .', 'Fig . 2 shows the compiled data from the first experimental protocol ( PROT1 ) [17] , reprocessed for comparison with the predictions of equation 27 .', 'The results of fitting equations 24 and 25 to the experimental data are also shown in Fig . 2 A and B . And Table 4 provides the values of the coefficients a and b of the curves fitting the power law ( equation 24 ) for each initial angle A0 .', 'In the log-log plot in Fig . 2 . A , the results of each individual experiment at a given A0 fit well to a straight line M ∼ I−b with values of b between 0 . 36 and 0 . 44 ( Table 4 ) .', 'Except in cases in which A0>90° , the coleoptile response is independent of the initial angle A0 , as predicted by the ARC model ( in Fig . 2 . A ) , and these data collapse on a single master curve .', 'On the semi-log plot , however , no convincing fit can be identified .', 'The power law relationship therefore seems to better describe the results than the logarithmic relationship does .', 'Under the power law dependency of M on light intensity I , the coefficient of determination was R2 ∼ 0 . 91 , such that our model captured 91% of the total changes in the steady-state tip angle attributable to changes in the initial angle A0 and fluence rate I .', 'The response for higher angles does not follow the first order relation expressed in the AC model anymore , but the sine laws .', 'It is then expected that , for inclination angles A0 > 90° , the experimental data are adrift from this fit and vary non-monotonously , as A varies in a way that can no longer be accounted for by the ARC model .', 'This is indeed observed in Fig . 2 .', 'The second protocol ( PROT2 ) provides a different way to measure the same parameter than PROT1 , even if the protocol is different .', 'The plot of AR/AP = A0/AP as a function of the experimental values of I should then collapse on the single master curve defined with PROT1 in Fig . 2 . C .', 'Again , the model predictions were fair at small angles , AR/AP ≤ 0 . 4 , but for larger angles the experimental data diverged non-monotonously from the model .', 'It is interesting to note that the results fit well to both the logarithmic law and the power law [17] .', 'Finally a last testable prediction of the ARC model is that the plot of AP/AR as a function of M , and hence of the light fluence rate I , should be S-shaped with two asymptotes:', 'i ) when the light intensity is low , gravity dominates and AR = 0 and', 'ii ) when the light intensity is high .', 'Moreover the equilibrium AR is shifted toward the orientation of incident light such that AR = AP ( Fig . 4 ) .', 'These predictions are also consistent with the experimental data shown in Fig . 2 . D .', 'Now that experimental agreement has been found with features of the photogravitropic models , it is interesting to discuss more in details the properties of the models and the insights it provides on the photo- and gravi-tropic control .', 'This will be conducted through considering variant models of increasing complexity .', 'In order to reach insights on the dynamic of the interactions between photo - , gravi- ( and proprio- ) -ceptions on the control of the tropic movement , we will consider the case of a tropic motion at constant incident light intensity I , therefore studying the ARC model ( the influence of I through changes in M , the ratio between photoception and graviception , having been clarifyied previously ) .'], ['This study shows that a minimal modeling approach used to study shoot gravitropism , which gave rise to the AC model [2] , can be readily extended to study the interactions among photoception , graviception and proprioception .', 'The extended model , the ARC model , is minimal but complete .', 'Simple dimensionless control parameters can be estimated from the ARC model , and the model is tractable and easy to understand .', 'Furthermore , it is possible and straightforward to include the dependency of photosensitivity on light intensity I , thereby producing an augmented ARC model .', 'Predictions from the latter model were assessed against detailed quantitative experimental results [17] .', 'The model agreed very well with the data for small initial angles .', 'All the experimental curves at various inclination angles and light intensities collapse to a single curve when using the ARC model with a power-law dependency between the photograviceptive control number M and the intensity of light I . Although the assessment is not comprehensive , the evidence presented supports the validity of the model .', 'The hypotheses that the action of the tropic motor is fully driven by perception-regulation processes ( H1 ) , that the angles between the gravity and the light fields are first-order variables influencing the tropic dynamical movement ( H2 ) , that the two types of perceptions act additively ( H4 ) , and that the balance between gravisensing and photosensing depends on the perception of light intensity ( H5 ) are upheld in the model .', 'The simplified versions and limiting cases of the ARC model served as useful tools for analyzing the dynamical interactions associated with perception during tropic movement , yielding five major insights .', 'i ) Despite having different set-points , gravitropism and phototropism act together to align the organ with the direction defined by AR ( equation 26 ) .', 'AR is determined by the dimensionless number M and hence by the ratio between gravisensitivity and photosensitivity .', 'As in the AC model , the tropic movement is controlled globally through the different types of tropic perception , which together drive the local curving velocity .', 'The final steady-state shape reflects the ratio of the respective sensitivities to light and gravity [2] .', 'This may seem to be at odds with the phenomenological model for PGEA proposed by Galland , summarized in equation 2 [17] , which states that the global steady-state shape at photo-gravitropic equilibrium depends additively on the phototropic and gravitropic stimuli , not on their ratio .', 'However this phenomenological model for PGEA becomes problematic when considering the limiting cases .', 'According to equation 2 , in darkness ( i . e . , when I = 0 ) the PGEA = −g sin ( A0 ) .', 'In other words , the gravitropic equilibrium orientation is a function of the initial angle .', 'However , when there is no light , the set-point becomes simply the gravitropic set-point angle ( GSA ) [28] , and the equilibrium shape is driven by the GSA and by the graviproprioceptive number B [2] .', 'For ortho-gravitropic organs ( i . e . , in which the GSA is vertical ) in plants in which graviception dominates proprioception ( where the value of B is large ) , experimental observations confirm that the equilibrium tip angle is indeed vertical , that is , PGEA = 0 for any initial angle A0 [2 , 22] .', 'The final gravitropic steady-state shape does not depend on sin ( A0 ) at all .', 'Rather , the sine law dependency applies to the transient rate of changes in curvature , not to the final steady-state shape [1 , 2 , 4 , 34] .', 'In the other limiting case , when photoception dominates gravisensing ( e . g . , when the light is very bright , in microgravity or in agravitropic mutants ) , the set-point angle is determined by the direction of the light alone .', 'The steady-state tip angle should therefore tend to AP .', 'The expression of the AR in equation 26 accommodates this condition , whereas the phenomenological model for PGEA in equation 2 predicts a tip angle that would depend on the light intensity with no saturation .', 'The phenomenological model for PGEA [17] is thus not consistent for these two limiting cases .', 'The expression of the AR as the ratio between the sensitivities overcomes these problems directly and is also consistent with a complete dynamical model , the ARC model .', 'AR is thus a better indicator than the previously defined PGEA [17] .', 'In order to to unify the notation , we propose to follow [26
['Introduction', 'Materials and Methods', 'Results', 'Discussion']
['Land plants rely mainly on gravitropism and phototropism to control their posture and spatial orientation .', 'In natural conditions , these two major tropisms act concurrently to create a photogravitropic equilibrium in the responsive organ .', 'Recently , a parsimonious model was developed that accurately predicted the complete gravitropic and proprioceptive control over the movement of different organs in different species in response to gravitational stimuli .', 'Here we show that the framework of this unifying graviproprioceptive model can be readily extended to include phototropism .', 'The interaction between gravitropism and phototropism results in an alignment of the apical part of the organ toward a photogravitropic set-point angle .', 'This angle is determined by a combination of the two directional stimuli , gravity and light , weighted by the ratio between the gravi- and photo-sensitivities of the plant organ .', 'In the model , two dimensionless numbers , the graviproprioceptive number B and the photograviceptive number M , control the dynamics and the shapes of the movement .', 'The extended model agrees well with two sets of detailed quantitative data on photogravitropic equilibrium in oat coleoptiles .', 'It is demonstrated that the influence of light intensity I can be included in the model in a power-law-dependent relationship M ( I ) .', 'The numbers B and M and the related photograviceptive number D are all quantitative genetic traits that can be measured in a straightforward manner , opening the way to the phenotyping of molecular and mechanical aspects of shoot tropism .']
['Although plants are mostly seen as static , they are constantly moving to adapt to changes in their stature and to their environment .', 'Gravity and light , among others , are major factors that sculpt the shapes of plants .', 'Plants tend to grow in the direction of the light to get access to their energy resource .', 'At the same time , however , they need to maintain their balance and control their posture .', 'In a recent study , we showed that postural control is regulated by two types of perception: graviception and proprioception .', 'We extend that study to include light perception in order to obtain a unified description of plant tropism .', 'As the system is highly dynamic , a model is required in order to evaluate hypotheses against experimental data .', 'Our results show that the direction of plant growth is determined by the combined influence of light and gravity , whereas postural control is still regulated by gravity and proprioception .']
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