A system for modelling cell-cell interactions during plant morphogenesis

Abstract

Background and aims: During the development of multicellular organisms, cells are capable of interacting with each other through a range of biological and physical mechanisms. A description of these networks of cell-cell interactions is essential for an understanding of how cellular activity is co-ordinated in regionalized functional entities such as tissues or organs. The difficulty of experimenting on living tissues has been a major limitation to describing such systems, and computer modelling appears particularly helpful to characterize the behaviour of multicellular systems. The experimental difficulties inherent to the multitude of parallel interactions that underlie cellular morphogenesis have led to the need for computer models.

Methods: A new generic model of plant cellular morphogenesis is described that expresses interactions amongst cellular entities explicitly: the plant is described as a multi-scale structure, and interactions between distinct entities is established through a topological neighbourhood. Tissues are represented as 2D biphasic systems where the cell wall responds to turgor pressure through a viscous yielding of the cell wall.

Key results: This principle was used in the development of the CellModeller software, a generic tool dedicated to the analysis and modelling of plant morphogenesis. The system was applied to three contrasting study cases illustrating genetic, hormonal and mechanical factors involved in plant morphogenesis.

Conclusions: Plant morphogenesis is fundamentally a cellular process and the CellModeller software, through its underlying generic model, provides an advanced research tool to analyse coupled physical and biological morphogenetic mechanisms.

Fig. 1.

(A) Plant cellular architecture can be broken down into entities at different scales, represented here on horizontal planes to describe plant structure and functioning. (B) The entities in each level of description establish interactions with other plant constituents, and it is possible to determine a topological neighbourhood for any entity: a cell is related to its neighbours according to h, but also within the organ it belongs to via v⁺ and the walls that define its boundaries via v⁻. (C) The evolution of such properties is relevant to autonomous processes, e.g. f₂¹ function in cell X₂¹, but is also due to interactions between entities in the topological neighbourhood, e.g. via f₃⁺, f_1– and f₂² functions. (D) Changes in the network of interactions are due to growth mechanisms and can be broken down into birth and death operators: the division of the cell X₂¹ results from the deletion of four walls, i.e. X¹₁, X^h1₁, X⁵₁, X^h5₁, and the creation of ten new walls (eight subdivision from previous walls plus the two new walls separating the newly created cell). Entities associated with new walls are then defined through the inheritance function g₁ and those associated with the two daughter cells through g₂.

Fig. 2.

Biomechanical model for cell expansion in morphogenesis: cell wall response to turgor pressure through a viscous yielding of the cell wall, compensated at the same time by thickening to maintain a constant cross-section.

Fig. 3.

(A) In the CellModeller software, a C++ chained data structure was developed in order to define explicitly the topological neighbourhood of any entity. This data structure is wrapped in a Python interface and can be used for various types of applications (extraction and reconstruction of real cellular architectures, statistical analyses, visualization and simulations). (B) Models can therefore be programmed by an association of self-contained scripts, operating on a type of entity and containing a description of both autonomous processes and interactions within the architecture. For any level of description, one script is used to define inheritance rules (function g_s), and the second script encodes the continuous dynamics of attributes (function f_s). The latter script also controls proliferation rules (division and death) and triggers architectural changes in the system.

Fig. 4.

The generation of the spacing patterns of trichomes in arabidopsis, results from a group of activators (in green) and inhibitors (in red). GL3 is a positive regulator of trichomes which associate with GL1 and TTG1 to form a complex that activates the trichome developmental genes which are reduced here to the single GL2 gene. In trichome cells, levels of the inhibitors increase and diffuse into neighbouring cells, where they block the activity of the activating complex, and in turn the trichome fate.

Fig. 5.

Patterns of gene expression obtained from the genetic regulatory network described in Fig. 4: (A) concentration of the GL1/TTG/GL3 complex; (B) concentration of TRY; (C) concentration of CPC; (D) concentration of GL2 in the cells (see Supplementary Information Video ‘Trichome-1’ available online).

Fig. 6.

The modelled genetic regulatory network described in Fig. 3 can be mutated by preventing expression of particular genes or by imposing a constant rate of synthesis in a particular gene product. Phenotypes of such mutated regulatory networks exhibited similar properties as real mutant as seen in the distribution of the GL1/TTG/GL3 complex: the wild-type phenotype showed sparse distribution in initiation sites (A), the CPC mutant phenotype had a higher density of trichome cells (B), the TRY mutant phenotype exhibited clusters of trichome (C) and gl3 overexpression increased the density of trichomes (D) (see Supplementary Information Video ‘Trichome-2’ available online).

Fig. 7.

In the primordia, influx and efflux carriers are thought to direct auxin flows and maintain regions of higher concentrations. The localization of the carrier molecules that maintain such patterns are inherent to feedbacks with the levels of auxin concentration and fluxes in cells. In AUX1 transport, the feedback F is inherent to the concentration of auxin in the cell. In PIN1 in the L1 layer, the rate of allocation is positively correlated with the concentration of auxin in the neighbouring cell. The canalization of the flux of auxin by PIN1 deeper in the tissue is induced by the flux of auxin through the cell wall.

Fig. 8.

Establishment of the ‘reverse fountain’ cycling of auxin: (A) initial conditions, (B) direction of flux towards the local maxima of auxin concentration; (C) redirection and canalization of the flux towards deeper tissues. The transport of auxin is mediated by carrier molecules distributed heterogeneously on cell walls: (D) AUX1 influx carrier conveys auxin to the L1 layer, (E) PIN1 efflux carrier directs the flux towards the site of maximum concentration in the L1 layer, (F) canalization process by PIN1 in deeper cells redirects the flux downwards (see Supplementary Information Video ‘Fountain’ available online).

Fig. 9.

The influence of mechanical interactions and tissue morphogenesis was illustrated by the simulation of an outgrowth generated by three tissues expanding at different rates. A fast-growing tissue (green) is adherent to two slowly growing surrounding tissues (blue, orange): (A–C) different stages of the development of the outgrowth; (D) strain rate distribution in cell walls (10⁻¹ h⁻¹); (E) areal strain rate (10⁻¹ h⁻¹) (see Supplementary Information Video ‘Outgrowth’ available online).