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Cell expansion as a mechano-hydraulic process: the Lockhart model
Turgor pressure (P ) is the force that water molecules exert per unit surface of the plasma membrane. Turgor pressure and mechanical properties of the wall have been traditionally related to cell expansion. Relaxation of the wall stress is crucial because it reduces water potential of the cell which, in turn, drives water influx. Water uptake occurs passively following the difference in water potential ∆Ψ = Ψoutside −Ψinside, being Ψ = P −π, the difference between turgor and osmotic (π) pressures. Movement of water across the membrane depends on the permeability of plasma membrane, L, which is a function of the physical structure of the membrane and the movement of water by aquaporins, and the surface area A [44]. Changes in water volume are given by dVdt = AL∆Ψ.
As water enters the cell, turgor pressure increases and when it surpases the critical threshold, called yield turgor (Y ), irreversible expansion of the cell wall occurs. This relationship is represented by: dVdt = φ(P − Y ).
with φ the extensibility of the cell wall, which can be defined as « the ability of the wall to irreversibly increase its surface area » [9]. Hence, cell expansion is a coordination between water uptake and yielding of the cell wall, a mechano-hydraulic process. These equations are known as Lockhart equations, and they describe growth of plant cells when pressure is constant, dPdt = 0 [13].
Types of cell wall expansion
Cell walls are inhomogeneous in their molecular architecture and thickness whereby they do not yield to turgor pressure uniformly on the surface. Cell walls expand by polymer creep, a process in which microfibrils and associated polysaccharides slide to irreversible expand the surface area of the wall [8]. It occurs when wall loosening mechanisms modify the molecular matrix of the wall and allow it to relax the mechanical stress created by turgor pressure and, therefore, yield to this force [8, 9]. The rate of wall loosening is related to the concept of wall extensibility φ, that we previously referred to. The spectrum of wall expansion types span from localized, as in tip-growing cells like pollen tubes, to diffuse, as in potato tube parenchyma [7]. Tip expansion occurs by apical cell wall synthesis and is commonly accompanied by parallel alignment of cortical microtubules with respect to the growth axis, zonation of organelles (mitochondria, Golgi and the endoplasmic reticulum) in the subapical region, and apical accumulation of vesicles (Figure 1.2). Diffuse expansion occurs when cell wall synthesis is distributed along the cell and cellulose microfibrils are randomly arranged, and it has no accumulation of vesicles nor organelles (Figure 1.2).
Although these have been commonly features associated to tip- and diffuse-growth, new discoveries suggest that this categorization might not be so clear-cut. Tip-growing trichomes of Arabidopsis had transversely aligned microtubules, as expected, but showed an apical region depleted from microtubules [26]. An actin meshwork operates in this microtubule-depleted region promoting diffuse growth and a wall thickness gradient that, according to model simulations, explains the tip bias to elongation and its rapid expansion [26]. This example suggests that the coordination of events to generate plant shapes might be more complex than usually assumed.
Cell wall loosening enzymes
Modifications of the wall that lead to stress relaxation can be caused by different types of signals such as hormones, e.g., auxin, and reactibe oxygen species (ROS). Some candidates of wall loosening enzymes include: expansins, xyloglucan endotransglycolase/hydrolase, and endo-(1,4)-β-d-glucanase. We will focus on the activities of expansins amd endo-(1,4)-β-d-glucanase because they have been studied in our model system, cotton fiber (subsection 1.3.2). Growing cell walls have a pH between 4.5 and 6 which typically activates expansins. Exogenous expansin application stimulates cellular growth and ectopic expression stimulates overall plant growth, while its gene silencing has the opposite effect [12, 36, 52]. Although the exact molecular action is unknown, expansins are believed to disrupt non-covalent bonds between cellulose microfibrils and between xyloglucan and cellulose [29]. The endo-(1,4)-β-d-glucanase activity has been correlated to fruit softening, absicion and growth. Their particular biochemical properties, substrate specificities and function in cell wall loosening are still unknown [8]. Potential substrates are cellulose and xyloglucan and indirect evidence suggests that they may loosen the wall by releasing xyloglucans from cellulose microfibrils [17, 34].
Fluxes of water
Movement of water molecules across the plasma membrane is made efficient through membrane wa-ter channel proteins, called aquaporins, and plasmodesmata. Based on sequence similarity aquaporins (AQPs) are divided into four subfamilies: plasma membrane intrinsic proteins (PIPs), tonoplast intrinsic proteins (TIPs), Nodulin26-like intrinsic proteins (NIPs), and small basic intrinsic proteins (SIPs) [22]. As indicated by the name, PIPs and TIPs localize at the plasma and tonoplast membranes, respectively. PIPs are further divided into PIP1 and PIP2, while TIPs fall into several classes (αTIPs, βTIPs, γTIPs, etc). Evidence linking AQPs to water transport comes from heterologous expression of AQPs in Xenopus laevis oocytes system which results in a considerable increase in the osmotic water membrane permeabil-ity (PF ) [27]. PF describes overall water movement in response to osmotic or turgor pressure gradients, ∆P and ∆π, respectively. Additionally, it was found through mercury inhibition of AQPs that PF of the membrane can be reduced by 70-80% in Chara and tonoplast vesicles from tobacco suspension cells [18, 28]. PF and L permeability parameters give estimations of the membrane water transport capacity and are related by : Pf = LRT /V , with R, the universal gas constant; T, absolute temperature, and V, water volume [27].
Biochemistry of the cell wall of cotton fiber during development
Cell wall remodelling takes place during expansion of cotton fibers. Components of primary cell wall are mostly synthesized during the expansion stage and it is considered that the thickness of the cell wall remains constant for at least 12 DAA [30]. Measurements of mRNAs by reverse transcription qPCR analysis revealed that mRNA levels of expansins and endo-1,4-β-glucanase are high at the begining of fiber development and decrease gradually as cell espansion stops [39, 43]. Synthesis of the primary cell wall is characterized by large amounts of xyloglucans which correlate with the rate of cell expansion, when the rate of expansion decreases xyloglucan amounts also decrease [43]. By 17-20 DAA secondary wall deposition starts and is characterized by a gradual increase in the level of cellulose content [30, 43]. At the end of the secondary wall synthesis stage, final thickness of the secondary cell wall can be 3 to 6 µm [16].
Aquaporins (AQPs)
Northern blot analysis and real-time qPCR demonstrated that AQPs of G. hirsutum, GhPIP1-2 and GhγTIP1 are expressed in cotton fiber cells during cell expansion having peak values at 5 DAA [25]. Another study found that four genes of GhPIP2 are specifically expressed in cotton fibers during growth [24]. Knockdown expression of four GhPIP2 genes by RNA interference (RNAi) reduced the final length of the fibers [24]. This down regulation did not affect density of fibers in the ovules nor the emergence of fiber initials but the rate of expansion of the fiber was much slower compared to wild type (WT) fibers [24]. Ligon-lintless-1 (Li1 ) and -2 (Li2 ) mutants are monogenic dominant mutants that have shorter fiber cells. Identification of common fiber elongation related genes in fibers of (Li1 ) and (Li2) found that aquaporins were among the most significantly down-regulated genes [31]. Measurements of sap osmolarities of these mutants revealed that osmotic pressure of Li1 fibers was significantly lower compared to WT fibers at 3-8 DAA [31]. Li2 mutant fibers also had lower osmotic pressure than WT fibers at 3-5 DAA but higher than WT at 24 DAA [31]. These findings suggest that AQPs are important regulators of cotton fiber cell expansion. However, no measurements of turgor pressure were performed in these mutants as a way to test the relationship between AQPs and turgor.
Sources of solutes
To maintain expansion for long periods, the fiber cell has to compensate dilution of solutes that occurs as water enters into the cell. Soluble sugars, K+ , and malic acid, account for ∼80% of sap osmolarity [11]. Malate is synthesized in the cytoplasm through fixation of CO2 by phosphoenolpyruvate carboxylase and sugars and K+ are imported from the seed coat phloem [1, 21]. Patterns of expression of sucrose and K+ membrane transporters – GhSUT1 and GhKT1 – were carried in G. hirsutum fibers. These analyzes found that both genes had low transcript levels at 6 DAA and increased to a maximum level at 10 DAA, whereupon GhSUT1 remained relatively high up to 16 DAA while GhKT1 decreased faster [39].
Another source of solutes that potentially contributes to avoid dilution is the Vacuolar Invertase 1 (VIN1). VIN1 is responsible for the hydrolization of sucrose into glucose and fructose and its activity has been linked to expanding tissues like sugar beet petioles [14]. In cotton fibers, its expression was found to be several times higher than in other tissues like roots, stems and leaves [50]. It was reported that VIN1 activity was higher at 5 and 10 DAA and lower at 20 DAA. Coincidently, the concentration of glucose and fructose was higher at 10 DAA and by 15 DAA it lowered. When VIN1 expression was silenced by means of RNAi construct, VIN1 activity was reduced by 33% and this resulted in shorter fiber cells [50].
Effects of parameter changes in the behavior of the system
To study the effects that parameters had on the behavior of cotton fiber growth, we performed two cases of sensitivity analysis around middle values: (a) effect of one parameter on the value of ratios when all other parameters take their middle values; (b) effect of one parameter on ratios when all other parameters span their ranges. For the second case, (b), we plotted the mean and standard deviation (SD). The complete list of plots can be found in the Supplementary Material 1. Accord-ing to these results, parameters can be classified in two groups: those that increase growth (α, Lr, φ and Pseed) and those that decrease it (µ, πseed, and Y) (Figure 2).
It has been shown that over expression of molecules that contribute to the ac-cumulation of solutes within cotton fibers, such as VIN1, can enhance the length of the fiber, probably by increasing the osmotic pressure (29). When more solutes are accumulated, water potential of the fiber lowers which drives water into the fiber that, in turn, increases turgor pressure. In our model, increases in the value of α result in larger πmax/πmin and Pmax/Pmin for case (a) which indicates that osmotic and turgor pressures have more changes during the simulation (Sup Mat 1.1). According to the value of VF/VI, the cell grows more with larger values of α, which seems to be in agreement with the experimental observations (Figure 2). Likewise, increasing the value of Lr results in larger cell volumes (Figure 2), which is consistent with experiments showing that overexpression of the plasma membrane aquaporin PIP2 in G. hirsutum results in longer fiber cells (19). As the wall ex-tensibility, φ, becomes larger, water potential decreases creating the conditions for water import. This situation is expected to result in larger cell volume which is the result recovered in our simulations (Figure 2a).
Influx of water and solutes through plasmodesmata channels is favoured when Pseed >Pfiber. When the value of Pseed is increased we observed that dπfibert3/α, the efflux of water and solutes decreases and it becomes positive, indicating that as Pseed becomes larger, water and solute molecules move through plasmodesmata into the fiber cell (Sup. Material 1.1). As expected from this shift, we observe that the fiber attains larger volumes (Figure 2a). Increasing the yield turgor has the opposite effect on fiber growth (Figure 2b) which may be explained by the fact that a larger Y means that turgor pressure has to reach larger values to make the cell growth.
The effects observed for the changes in parameters on volume of the cotton fiber follow a similar trend between cases (a) and (b), however, in case (b) the changes in fiber volume seem to be much larger than in (a).
Table of contents :
1 Introduction
1.1 The plant cell wall
1.1.1 Cell wall components
1.2 Cell expansion as a mechano-hydraulic process: the Lockhart model
1.2.1 Types of cell wall expansion
1.2.2 Cell wall loosening enzymes
1.2.3 Fluxes of water
1.3 Cotton fiber as a study system
1.3.1 The pattern of cotton fiber expansion
1.3.2 Biochemistry of the cell wall of cotton fiber during development
1.3.3 Aquaporins (AQPs)
1.3.4 Sources of solutes
1.3.5 Plasmodesmal permeability
1.4 General questions
2 Interplay between turgor pressure and plasmodesmata during plant development
3 A mechano-hydraulic model for the study of cotton fiber elongation
4 Discussion
4.1 Regulation of plasmodesmal permeability might have been critical for the evolution of complex multicellular plants
4.2 Turgor may affect plasmodesmal permeability
4.3 Plasmodesmal permeability may affect turgor-driven growth of cotton fiber cells
4.4 Removing the dynamics of plasmodesmal permeability may preclude non monotonic behavior of turgor pressure
4.5 General Conclusion
5 Appendix
