The brown algal mode of tip growth: Keeping stress under control

Tip growth has been studied in pollen tubes, root hairs, and fungal and oomycete hyphae and is the most widely distributed unidirectional growth process on the planet. It ensures spatial colonization, nutrient predation, fertilization, and symbiosis with growth speeds of up to 800 μm h−1. Although turgor-driven growth is intuitively conceivable, a closer examination of the physical processes at work in tip growth raises a paradox: growth occurs where biophysical forces are low, because of the increase in curvature in the tip. All tip-growing cells studied so far rely on the modulation of cell wall extensibility via the polarized excretion of cell wall–loosening compounds at the tip. Here, we used a series of quantitative measurements at the cellular level and a biophysical simulation approach to show that the brown alga Ectocarpus has an original tip-growth mechanism. In this alga, the establishment of a steep gradient in cell wall thickness can compensate for the variation in tip curvature, thereby modulating wall stress within the tip cell. Bootstrap analyses support the robustness of the process, and experiments with fluorescence recovery after photobleaching (FRAP) confirmed the active vesicle trafficking in the shanks of the apical cell, as inferred from the model. In response to auxin, biophysical measurements change in agreement with the model. Although we cannot strictly exclude the involvement of a gradient in mechanical properties in Ectocarpus morphogenesis, the viscoplastic model of cell wall mechanics strongly suggests that brown algae have evolved an alternative strategy of tip growth. This strategy is largely based on the control of cell wall thickness rather than fluctuations in cell wall mechanical properties.

Graphic representation of the Lockhart equation (S1).

Viscoplastic model of the cell wall
The viscoplastic model shows how the cell wall physical properties at microscopic level drive cell growth. This model focuses on the case of the apical cell in a filamentous organism. The cell is mostly cylindrical, with a dome-like tip, which is responsible for growth. Following Dumais et al. [3], this section considers an elementary portion of cell wall. Turgor pressure, acting on the whole cell wall, results in a local wall stress σ e , which can vary from point to point. As the cell is supposed to admit a circular symmetry, σ e only depends on the curvilinear abscissa s. This is the case for all variables, which should therefore be denoted as functions of s, but this will be usually omitted to keep notation simple. The small cell wall fragment expands if σ e > σ y , where σ y is the yield threshold [4]. σ y is a local cell wall property, which plays the same role as Y in the Lockhart equation (S1).
The stress can be partitioned into three directions: meridional (s), circumferential (θ) ) and normal (n). As the cell wall is thin compared to the cell dimensions, the normal component of the stress is considered negligible beside the two others [5]. Knowing the turgor pressure P, and having measured for each abscissa the cell wall thickness δ and the curvatures κ s and κ θ) (Fig B), the local components of the stress are [6]:

Transverse isotropy
If the cell wall is transversely isotropic (i.e. the cell wall properties are identical in the meridional and circumferential directions), then the resulting value of global stress is: where ν is a dimensionless value denoting the flow coupling, equivalent to the Poisson ratio.
In these conditions, the cell wall part is subject to a strain whose rate is ε e = 1 L dL dt .
Plasticity is characterized by the relation between stress and strain rate: The global strain rate decomposes into its components according to: with: In these equations, Φ represents the local cell wall extensibility, which is the microscopic equivalent to the macroscopic extensibility Φ L in Lockart's equation (S1). The meridional and circumferential components of the strain rate induce an increase in cell wall surface, which is responsible for cell growth. The distribution of this increase between the two directions defines the local changes in cell shape which, taken together in the whole cell come out as global shape stability.

Orthogonal growth
If cell wall growth is orthogonal (i.e. each point of the wall moves perpendicular to the tangent to cell wall), then partitioning of growth between the meridional and circumferential axes becomes a direct consequence of cell shape: The flow coupling also illustrates this property: At the tip of the dome, ν = 0 while in the cylindrical part, ν = 1/2 .
In these conditions, the velocity of a point of the cell wall is: The viscoplastic model thus allows to understand the growth process at cell level as the integration of local strains resulting from geometrical and physical parameters, acting on infinitesimal parts of the cell wall.

Expected velocity and expected strain rate
Within the frame of the viscoplastic model, it is possible to set functions Φ(s) and σ y (s) which maintain insofar as possible the cell morphology during growth.
The target behavior of an apical cell development model is to grow without morphological change (Fig C). Locally, the dome is constantly reshaped, but at steady state the global result of the growth process is similar to a uniform translation along the longitudinal axis. Actually, under the hypothesis of orthogonal growth, each point of the dome moves in a direction that is normal to the cell wall. Over time, the local Fig C: a, A dome-like apex grows due to expansion of the cell wall in the dome region. b, Provided that this process maintains cell shape, it is globally equivalent to a uniform translation of the dome (self-similar tip growth, [7]), where the cylindrical part of the cell is considered of infinite length. c, For any point, the direction of the actual instantaneous velocity is normal to the cell wall, and its modulus is constrained by the necessity to maintain the global morphology of the cell.
As ε θ =κ θ V n (eq. S9), rearranging (S5) and (S7) allows to express the local contribution to the cell deformation in the form of an expected strain rate: At each point (that is: for each value of s), the values of κ θ) , V n , σ s , σ θ) , σ e , ν and K are known: they can be deduced from the cell shape, cell wall width and turgor pressure (see above in the same paragraph, and equations S2, S3, S5, S6, S8). Therefore, ε * ( s) is computed without any knowledge about Φ(s) and σ y (s).

From expected strain rate to plasticity parameters
Under the assumptions of the viscoplastic model, the cell deformation results from local strain, i.e. the expected strain rate is nothing but the Lockhart strain rate, or: Let us explicitly express the relation between stress and strain rate (equation S4) as functions of s:ε e (s)=Φ( s)(σ e (s )−σ y (s)) (S12)