# Models to reconcile plant science and stochasticity

Sam Collaudin, Vincent Mirabet

► To cite this version:

Sam Collaudin, Vincent Mirabet. Models to reconcile plant science and stochasticity. Frontiers in Plant Science, 2014, 5, pp.1-4. 10.3389/fpls.2014.00643. hal-02629980

HAL Id: hal-02629980

https://hal.inrae.fr/hal-02629980

Submitted on 27 May 2020

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# Models to reconcile plant science and stochasticity

Sam Collaudin<fnref n="1" />,<fnref n="2" /> and Vincent Mirabet<fnref n="1" />,<fnref n="2" />*

**Edited by:**

Naomi Nakayama, University of Edinburgh, UK

**Reviewed by:**

James Locke, University of Cambridge, UK

Jose Teles, University of Cambridge, UK

**Keywords:** stochasticity, plant development, modeling, pattern, morphogenesis, tissue

Plants are modular organisms that exhibit diverse adaptations to variability. This variability can be intrinsic in nature, as in the case of cell shape or division plane stochasticity, protein distribution in a cell, variations in internal mechanical properties etc... (Altschuler et al., 2008; Besson and Dumais, 2011). It can also be extrinsic, as with variations in environmental conditions at different time scales (Wolpert et al., 1998; Sultan, 2000; Franklin, 2009; Leyser and Day, 2009). When it comes to rationalizing data acquisition and interpretation, one has the tendency to define what part of the variability is arguably unhelpful stochasticity and what part does in fact contain meaningful information.

Systems biology, which combines methodologies from various disciplines, can be used to understand the mechanisms of development. For example, complex network analysis (Lucas et al., 2011), computer simulations (Band et al., 2012) or physical measurements through atomic force microscopy (Milani et al., 2014) can be combined with biological experiments. For instance, such an approach has been able to produce reasonable explanations for how patterning at the meristem level can lead to the stem structure (Prusinkiewicz et al., 1995). Stochasticity in models as a variable or as a methodological tool has been a subject of interest for many years in physics and mathematics (Saguès et al., 2007; Friedrich et al., 2011; Wilkinson, 2011). Studies have already been published in biology but only a few focused on plant development, and are often more recent (for a review of this aspect, see Meyer and Roeder, 2014). Along with a better

understanding of growth processes, those studies have also illustrated how our vision of stochasticity was previously too derogatory (Kliebenstein, 2012). Those new methodologies illustrate how stochasticity can be both a consequence and an origin of core mechanisms in development.

Here we use specific examples to illustrate how mathematical or computational models are well-suited to the study of stochasticity in plant functions. Moreover, models enable the use of measured phenotypic stochasticity at multiple scales to elucidate the underlying processes. We suggest that models used for such purposes do not need to be overly complex, and various complex models of the same process will in fact converge toward similar conclusions. We will focus our attention on apical meristems and the growth that they generate, where cell-cell interactions underlie the emergence of various interesting properties of the tissues and organs.

## STOCHASTICITY CAN BE BUFFERED BY GENETIC NETWORKS AT CELLULAR AND TISSULAR SCALES

It has been shown that low levels of a protein or a chemical component induce a high level of noise that can impact on pattern formation (Shnerb et al., 2000). For instance, in *Drosophila*, the *Hunchback* (*Hb*) gene is crucial for the proper segmentation of the embryo, and is regulated by *Bicoid* (*Bcd*). The Bcd gradient can cause intense noise due to the small numbers of molecules. Nevertheless, the definition of segmentation and boundary position are well-conserved between embryos despite Bcd stochasticity. Holloway proposed in his study that noise can be affected by the

number and the strength of binding sites on a promoter (Holloway et al., 2011). He develops computational models to confirm that the high number of Bcd binding sites on the Hb promoter reduce stochastic noise of Bcd gradient.

Complex gene regulatory networks drive morphogenesis in all multicellular contexts. Feedback and redundancy within these networks compensate for intrinsic or extrinsic stochasticity. For instance, flower formation is driven by the activation of a small network mainly composed by *LEAFY* (*LFY*), *APETALA1* (*AP1*), *CAULIFLOWER* (*CAL*), and *TERMINAL FLOWER1* (*TFL1*). This network is regulated by environmental and physiological inputs to start flower initiation at the appropriate time for reproduction. This network contains many feedback loops and mutual activations, such as the induction by *LFY* of *AP1* and *CAL*, which themselves positively regulate *LFY*. These interactions can buffer the environmental noise to obtain the formation of a robust pattern and to avoid the reversal of flowering (Blazquez et al., 2006).

## MODELS EXPLAIN THE ROBUSTNESS OF PATTERNING

Plant tissues, even those as little differentiated as meristems, exhibit strong self-organizational properties. Intense local cell-cell interactions through diverse exchanges (Murray et al., 2012; Landrein and Vernoux, 2014) contribute to the emergence of patterns. Those patterns may be in the form of either simple genetic differentiation or more complex morphogenetic events. Among the various properties of such self-organized

<footnote n="1">¹ Reproduction et Développement des Plantes, INRA, CNRS, Ecole Normale Supérieure de Lyon, Université Claude Bernard Lyon 1, Lyon, France</footnote>

<footnote n="2">² Laboratoire Joliot-Curie, CNRS, Ecole Normale Supérieure de Lyon, Lyon, France</footnote>

<footnote>*Correspondence: vincent.mirabet@ens-lyon.fr</footnote>
patterning, robustness is crucial for the
principal meristematic properties: their
ability for self-renewal or to produce vari-
ous lateral organs. Models are well-suited
to predict how a set of linked cells can
generate shape and differentiate following
emergent processes.

Auxin signaling processes are amongst
the best-studied cases of tissue pattern-
ing in plants, and furthermore, they have
also been extensively modeled and linked
to numerous biological observations. In
the auxin flux models, simple cell-to-
cell communication occurs via the local
amplification of auxin flux by the PIN-
FORMED proteins. The auxin response
patterns observed in the meristematic tis-
sues of the stem and at the vascular gen-
eration zones, are an emergent property
of those molecular interactions (Sassi and
Vernoux, 2013). This system clearly illus-
trates the robustness of emergent patterns
to external noise. External noise can be
as intense as multicellular injuries. It has
been shown that an injury to the meris-
tem can be compensated. The patterning
system is robust enough and can main-
tain the activity of organogenesis (Snow
and Snow, 1932; Reinhardt et al., 2005).
Computational Models of phyllotaxis can
predict how the plant might cope with
such ablations; the pattern is very quickly
deformed around the ablation, but re-
emerges naturally as growth continues
to produce healthy new cells. This resis-
tance to local injuries is also observed in
vasculature development. Cutting a part
of the provasculature induces its sponta-
neous reconfiguration, such that the new
vasculature is reshaped around the abla-
tion (Sauer et al., 2006). Models predict
that such reconfiguration does not need
any specific change in cell behavior, and
that cell-cell communication itself is suf-
ficient to enable such changes (Wabnik
et al., 2010).

STOCHASTICITY IN PATTERNING IS
FILTERED IN PLANT TISSUES

If spatial robustness is a natural outcome
of the self-organization described above,
this kind of patterning is itself some-
times stochastic. The phyllotactic angle
between successive lateral organs forming
on the shoot apical meristem (SAM) is
approximately 137 degrees in *Arabidopsis*
*thaliana*. For a long time, research has

focused on predicting mechanisms behind
the astonishing regularity of phyllotaxis
in various plant systems (Adler, 1997).
More recently, the close examination of
plants phyllotaxis has led to the discov-
ery of strange phenotypic alterations. The
*histidine phosphotransfer protein 6 (ahp6)*
mutant presents curious alterations called
M-shaped successions, where three succes-
sive lateral organs display altered angles
(Besnard et al., 2013). Other, more com-
plex, successions are visible with lower
frequencies. Strikingly, these types of alter-
ations also occur in wild type plants,
though less frequently. In order to under-
stand the source of this stochasticity and
the specific pattern of alterations, both sta-
tistical (Refahi et al., 2011; Guédon et al.,
2013) and agent (Mirabet et al., 2012)
models have been used to study phyl-
lotaxis in *Arabidopsis*. These models have
predicted that the auxin system, under
the influence of stochasticity, can sponta-
neously generate the alterations seen along
the stem. Indeed in mutants some organs
can be generated simultaneously, whereas
a delay (called the plastochron) occurs in
a typical normal situation. Because the
organs appear simultaneously, the way
they are arranged along the stem may be
inverted, thus producing the characteristic
M-shaped structure. This stochasticity in
timing would thus appear to be a sponta-
neous outcome of the spatial patterning of
the auxin system.

These studies have helped clarify that
the *Arabidopsis* SAM possesses a second
patterning system, based on the AHP6
protein, that partially overlaps the auxin
system and ensures that new primordia
will emerge successively through time.
Thus, the temporal stochasticity of the
auxin system is compensated for by a
second patterning process that filters it.
Without the use of a “systemic” view of
the entire patterning process, it would have
been difficult to decipher the role of the
AHP6 system.

STOCHASTICITY AS A SOURCE OF
PATTERNING AND MORPHOGENESIS

In developmental biology, stochastic gene
expression can lead to the formation of
coherent patterns. An example is in the
ommatidium of the *Drosophila* eye, which
consists of eight photoreceptor cells. Two
of them (R7 and R8) express rhodopsin,

which is responsible for the detection of
color. It has been shown that the sepa-
ration of “yellow” and “pale” ommatidia
determined by rhodopsin regulation in R7
and R8 is due to the stochastic expression
of the *SPINELESS* receptor (Wernet et al.,
2006). This stochasticity is both neces-
sary and sufficient for proper ommatidial
development. In this example, stochas-
tic gene expression at the cell level can
become instructional at the tissue level.

Through the use of simple activator-
inhibitor model systems, Turing managed
to describe the self-organization of vari-
ous spatial patterns (Turing, 1952). These
patterns mainly depend on the strength of
molecular interactions and on the geom-
etry of the domains where the activators
and inhibitors are expressed. In these com-
putational models, stochasticity is neces-
sary to trigger the dynamics that leads
to the final stable pattern. Stochasticity
of cell behaviors becomes the motor of
patterning. Nevertheless, this stochastic-
ity is in a way buffered by the interac-
tions, as its intensity has only a small
effect on the final pattern. In plants, exam-
ples of such systems exist in trichomes
positioning in leaves (Benítez et al., 2007;
Greese et al., 2012). Interactions can be
summarized into an activator complex
(that consists of WEREWOLF, GLABRA1,
GLABRA3, ENHANCER OF GLABRA3,
and TRANSPARENT TESTA GLABRA) as
well as the redundant inhibitory activ-
ity of CAPRICE and TRIPTYCHON. With
Turing-like models applied to those com-
ponents, the authors were able to repro-
duce the experimentally observed patterns.

Stochasticity is present not only in
gene expression, but is an inherent prop-
erty of cells, notably with respect to cell
growth. A recent study showed that cells
are able to interact mechanically to adapt
their growth depending on the behav-
iors of their neighbors (Uyttewaal et al.,
2012). Interestingly, this function seems
to increase variability instead of compen-
sating for it. In turn this positive feed-
back is necessary for correct morphogen-
esis of new primordia. Models predict
that an optimum exists between variabil-
ity of cell growth and feedback between
cells. Depending upon the relative strength
of both parameters, the tissue can grow
more or less efficiently. This intricate inter-
play between stochasticity and cell-cell
communication is a fundamental aspect of tissue morphogenesis, and would appear to be regulated. Models can help predict the optimal ratio between stochasticity and feedback necessary for proper morphogenesis. Interestingly, it is not this theoretical optimum that seems to be generated in meristems, a fact that may allow the tissue to undergo growth bursts, which may in turn lead to primordia emergence (Alim et al., 2012).

## SIMPLE MODELS TRANSLATE VARIABLE PHENOTYPES INTO VALUABLE INFORMATION

Complex systems can be modeled quite simply. An example is human crowds being modeled as simple interacting agents with very basic properties. Such models can efficiently predict the behavior of these groups (Helbing et al., 2000). Similarly, plant cells and tissues can also be modeled using such approaches. With a simple model such as that of Turing (with less than 10 parameters), it is possible to add noise measured at the cell scale, and study its consequences at an higher (tissue or plant) level. Thus, phenotypic variability at this higher level can be interpreted through the model, that gives the ability to search for the cellular parameters leading to the mutant phenotype of interest.

In the example of phyllotaxis described above, the types and frequencies of alterations may be interpreted through the use of the model. They are predicted to be an outcome either of alterations of the meristem structure or the auxin system. This scenario may be easily tested with further experimentation, for example searching for defects in the pin network or meristematic size.

This reasoning is in fact multiscale, each conclusion at one scale providing the data for models that focus on the link to the next level of organization. Our ability to measure variability and stochasticity at various scales has recently been increased. Experimental techniques and analysis tools have immensely improved the precision of measurements both spatially and temporally, and at various scales (Fernandez et al., 2010; de Reuille et al., 2014). Those new techniques point out the importance of heterogeneity and stochasticity in biological systems. Modeling approaches will be more and more helpful

in this new context to explain those data.

It is time to switch from seeing biology as clockwork perfection to looking at its natural variations more thoroughly. That will undoubtedly help us decipher where plants' real beauty is hidden: behind those so-called imperfections.

## ACKNOWLEDGMENTS

We thank Pradeep Das and Arezki Boudaoud for fruitful comments.

## REFERENCES

* Adler, I. (1997). A history of the study of phyllotaxis. *Ann. Bot.* 80, 231–244. doi: 10.1006/anbo.1997.0422

* Alim, K., Hamant, O., and Boudaoud, A. (2012). Regulatory role of cell division rules on tissue growth heterogeneity. *Front. Plant Sci.* 3:174. doi: 10.3389/fpls.2012.00174

* Altschuler, S. J., Angenent, S. B., Wang, Y., and Wu, L. F. (2008). On the spontaneous emergence of cell polarity. *Nature* 454, 886–889. doi: 10.1038/nature07119

* Band, L. R., Fozard, J. A., Godin, C., Jensen, O., Pridmore, T., Bennett, M. J., et al. (2012). Multiscale systems analysis of root growth and development: modeling beyond the network and cellular scales. *Plant Cell* 24, 3892–3906. doi: 10.1105/tpc.112.101550

* Benítez, M., Espinosa-Soto, C., Padilla-Longoria, P., Diaz, J., and Alvarez-Buylla, E. R. (2007). Equivalent genetic regulatory networks in different contexts recover contrasting spatial cell patterns that resemble those in Arabidopsis root and leaf epidermis: a dynamic model. *Int. J. Dev. Biol.* 51, 139–155. doi: 10.1387/ijdb.062183mb

* Besnard, F., Refahi, Y., Morin, V., Marteaux, B., Brunoud, G., Chambrier, P., et al. (2013). Cytokinin signalling inhibitory fields provide robustness to phyllotaxis. *Nature* 505, 417–421. doi: 10.1038/nature12791

* Besson, S., and Dumais, J. (2011). Universal rule for the symmetric division of plant cells. *Proc. Natl. Acad. Sci. U.S.A.* 108, 6294–6299. doi: 10.1073/pnas.1011866108

* Blazquez, M., Ferrandiz, C., Madueno, F., and Parcy, F. (2006). How floral meristems are built. *Plant Mol. Biol.* 60, 855–870. doi: 10.1007/s11103-006-0013-z

* de Reuille, P. B., Robinson, S., and Smith, R. S. (2014). Quantifying cell shape and gene expression in the shoot apical meristem using MorphoGraphX. *Methods Mol. Biol.* 1080, 121–134. doi: 10.1007/978-1-62703-643-6_10

* Fernandez, R., Das, P., Mirabet, V., Moscardi, E., Traas, J., Verdeil, J.-L., et al. (2010). Imaging plant growth in 4D: robust tissue reconstruction and line-eaging at cell resolution. *Nat. Methods* 7, 547–553. doi: 10.1038/nmeth.1472

* Franklin, K. A. (2009). Light and temperature signal crosstalk in plant development. *Curr. Opin. Plant Biol.* 12, 63–68. doi: 10.1016/j.pbi.2008.09.007

* Friedrich, R., Peinke, J., Sahimi, M., and Reza Rahimi Tabar, M. (2011). Approaching complexity by stochastic methods: from biological systems

to turbulence. *Phys. Rep.* 506, 87–162. doi: 10.1016/j.physrep.2011.05.003

* Greese, B., Wester, K., Bensch, R., Ronneberger, O., Timmer, J., Huulskamp, M., et al. (2012). Influence of cell-to-cell variability on spatial pattern formation. *IET Syst. Biol.* 6, 143–153. doi: 10.1049/iet-syb.2011.0050

* Guédon, Y., Refahi, Y., Besnard, F., Farcot, E., Godin, C., and Vernoux, T. (2013). Pattern identification and characterization reveal permutations of organs as a key genetically controlled property of post-meristematic phyllotaxis. *J. Theor. Biol.* 338, 94–110. doi: 10.1016/j.jtbi.2013.07.026

* Helbing, D., Farkas, I., and Vicsek, T. (2000). Simulating dynamical features of escape panic. *Nature* 407, 487–490. doi: 10.1038/35035023

* Holloway, D. M., Lopes, F. J., da Fontoura Costa, L., Travençolo, B. A., Golyandina, N., Usevich, K., et al. (2011). Gene expression noise in spatial patterning: hunchback promoter structure affects noise amplitude and distribution in *Drosophila* segmentation. *PLoS Comput. Biol.* 7:e1001069. doi: 10.1371/journal.pcbi.1001069

* Kliebenstein, D. J. (2012). Model misinterpretation within biology: phenotypes, statistics, networks, and inference. *Front. Plant Sci.* 3:13 doi: 10.3389/fpls.2012.00013

* Landrein, B., and Vernoux, T. (2014). "Auxin, chief architect of the shoot apex," in Auxin and its Role in Plant Development, eds E. Zažímalová, J. Petrášek, and E. Benková (Wien: Springer-Verlag), 191–212. doi: 10.1007/978-3-7091-1526-8_10

* Leyser, O., and Day, S. (2009). *Mechanisms in Plant Development*. Oxford: John Wiley & Sons.

* Lucas, M., Laplaze, L., and Bennett, M. J. (2011). Plant systems biology: network matters. *Plant Cell Environ.* 34, 535–553. doi: 10.1111/j.1365-3040.2010.02273.x

* Meyer, H. M., and Roeder, A. H. K. (2014). Stochasticity in plant cellular growth and patterning. *Front. Plant Sci.* 5:420. doi: 10.3389/fpls.2014.00420

* Milani, P., Mirabet, V., Cellier, C., Rozier, F., Hamant, O., Das, P., et al. (2014). Matching patterns of gene expression to mechanical stiffness at cell resolution through quantitative tandem epifluorescence and nanoindentation. *Plant Physiol.* 165, 1399–1408. doi: 10.1104/pp.114.237115

* Mirabet, V., Besnard, F., Vernoux, T., and Boudaoud, A. (2012). Noise and robustness in phyllotaxis. *PLoS Comput. Biol.* 8:e1002389. doi: 10.1371/journal.pcbi.1002389

* Murray, J. A. H., Jones, A., Godin, C., and Traas, J. (2012). Systems analysis of shoot apical meristem growth and development: integrating hormonal and mechanical signaling. *Plant Cell* 24, 3907–3919. doi: 10.1105/tpc.112.102194

* Prusinkiewicz, P., Hammel, M., Mech, R., and Hanan, J. (1995). "The artificial life of plants," in *Artificial Life for Graphics, Animation, and Virtual Reality (ACM SIGGRAPH)*, 7, 1–1, 1–38.

* Refahi, Y., Farcot, E., Guédon, Y., Besnard, F., Vernoux, T., and Godin, C. (2011). "A combinatorial model of phyllotaxis perturbations in *Arabidopsis thaliana*," in *Combinatorial Pattern Matching*, eds R. Giancarlo and G. Manzini (Berlin: Heidelberg: Springer), 323–335. doi: 10.1007/978-3-642-21458-5_28
Reinhardt, D., Frenz, M., Mandel, T., and Kuhlemeier, C. (2005). Microsurgical and laser ablation analysis of leaf positioning and dorsoventral patterning in tomato. *Development* 132, 15–26. doi: 10.1242/dev.01544

Saguès, F., Sancho, J., and García-Ojalvo, J. (2007). Spatiotemporal order out of noise. *Rev. Mod. Phys.* 79, 829–882. doi: 10.1103/RevModPhys.79.829

Sassi, M., and Vernoux, T. (2013). Auxin and self-organization at the shoot apical meristem. *J. Exp. Bot.* 64, 2579–2592. doi: 10.1093/jxb/ert101

Sauer, M., Balla, J., Luschnig, C., Wisniewska, J., Reinöhl, V., Friml, J., et al. (2006). Canalization of auxin flow by Aux/IAA-ARF-dependent feedback regulation of PIN polarity. *Genes Dev.* 20, 2902–2911. doi: 10.1101/gad.390806

Shnerb, N. M., Louzoun, Y., Bettelheim, E., and Solomon, S. (2000). The importance of being discrete: life always wins on the surface. *Proc. Natl. Acad. Sci. U.S.A.* 97, 10322–10324. doi: 10.1073/pnas.180263697

Snow, M., and Snow, R. (1932). Experiments on phylloaxis. II. The effect of displacing a primordium. *Phil. Trans. R. Soc. Lond.* B 222, 353–400. doi: 10.1098/rstb.1932.0019

Sultan, S. E. (2000). Phenotypic plasticity for plant development, function and life history. *Trends Plant Sci.* 5, 537–542. doi: 10.1016/S1360-1385(00)01797-0

Turing, A. M. (1952). The chemical basis of morphogenesis. *Phil. Trans. R. Soc. Lond.* B 237, 37–72. doi: 10.1098/rstb.1952.0012

Uyttewaal, M., Burian, A., Alim, K., Landrein, B., Borowska-Wykrt, D., Dedieu, A., et al. (2012). Mechanical stress acts via Katanin to amplify differences in growth rate between adjacent cells in Arabidopsis. *Cell* 149, 439–451. doi: 10.1016/j.cell.2012.02.048

Wabnik, K., Kleine-Vehn, J., Balla, J., Sauer, M., Naramoto, S., Reinöhl, V., et al. (2010). Emergence of tissue polarization from synergy of intracellular and extracellular auxin signaling. *Mol. Syst. Biol.* 6, 447. doi: 10.1038/msb.2010.103

Wernet, M. F., Mazzoni, E. O., Çelik, A., Duncan, D. M., Duncan, I., and Desplan, C. (2006). Stochastic spinless expression creates the retinal mosaic for colour vision. *Nature* 440, 174–180. doi: 10.1038/nature04615

Wilkinson, D. J. (2011). *Stochastic Modelling for Systems Biology*. Boca Raton, FL: CRC Press.

Wolpert, L., Beddington, R., Brockes, J., Jessell, T., Lawrence, P., and Meyerowitz, E. (1998). *Principles of Development*. London: Current Biology Ltd.

**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Received: 31 July 2014; accepted: 30 October 2014; published online: 14 November 2014.

Citation: Collaudin S and Mirabet V (2014) Models to reconcile plant science and stochasticity. *Front. Plant Sci.* 5:643. doi: 10.3389/fpls.2014.00643

This article was submitted to Plant Evolution and Development, a section of the journal *Frontiers in Plant Science*.

Copyright © 2014 Collaudin and Mirabet. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice.

No use, distribution or reproduction is permitted which does not comply with these terms.