# 4.3.3 Application of water relations equations to growing cells

While we have made a case for wall relaxation as a primary influence on cell expansion, there is also a requirement for cells to maintain sufﬁcient *P* in order to keep deforming walls and sustain growth. This is illustrated by the ‘pressure-block’ experiments (Figure 4.20) in which increasing pressures are required to block growth by lowering *P*. Under con-ditions of adequate water and solute supply, yielding charac-teristics of the cell wall (wall rheology) are major determinants of growth. However, in drought, severely drying winds or dehydrating, saline soils, water deﬁcits might be severe enough to cause complete loss of *P*. It is therefore important to relate variables which play a part in cell expansion, such as cell wall properties and water flow, to Equation 4.9.

Long-distance vascular transport in plants is driven by pressure gradients which arise either osmotically (such as in the roots of guttating plants) or hydraulically (transpiring canopies). Water enters the cells of growing tissues at a rate determined by local Y gradients and resistances. Hence, *P*, P, hydraulic conductivity (*L*_{p}) of the cell wall and plasma mem-brane and wall rheology will all influence cell growth rate. While some of these variables are difﬁcult to measure on a cellular scale, reasonable theoretical models have been devel-oped to describe how they relate to each other. The analysis to follow mostly relates to water flow into individual growing cells. However, the analogy to whole tissues is often apparent and will sometimes be considered where the two have principles in common.

How readily water flows through cells and tissues is encapsulated in the term hydraulic conductivity (displacement of water per unit of pressure and time). Hydraulic con-ductivity is especially important when water travels over long distances, such as across a stem, and where there are barriers to water flow, such as suberised layers of roots and tyloses in trees. The equation for water uptake (increase in cell volume) can be expressed as:

where is the rate of cell expansion (m^{3} s^{–1}), *L*_{p} is specific hydraulic conductivity (m Pa^{-1} s^{-1}), *A* is membrane surface area (m^{2}), *L* is hydraulic conductance (m^{3} Pa^{-1} s^{-1}) and ΔΨ is the gradient in water potential (Pa) generated by osmotic (ΔΠ) and turgor (ΔP) pressure gradients across membranes. Therefore, the rate of water uptake into cells is dependent on *L*_{p} and the inward gradient in Ψ. However, the delivery of water into growing cells through theri walls and membranes is only part of the growth process; we must also incorporate the characteristics of irreversible cell wall expansion into an analysis of growth, thus recognising the pivotal role of wall relaxation in growth.

Lockhart (1965) described cell expansion with a form of the following equation for *P*-driven growth:

where φ is the wall yielding coefﬁcient, sometimes referred to as wall extensibility (m^{3} Pa^{–1} s^{–1}) and *P*_{th} is threshold turgor pressure or yield threshold (m^{3} Pa). This equation and variations on it have become a paradigm for cell and tissue growth.

Equations 4.10 and 4.11 both provide plausible views of growth but raise the question of whether supply of water for cell expansion or irreversible yielding of cell walls is the key control step in growth. Experiments of Cosgrove cited above illustrate that relaxation of cell walls occurs independently of water inflow. It is generally considered that entry of water into rapidly growing cells is not a limiting factor for cell expansion. That is, Equation 4.11, which has no hydraulic term, is very relevant to growing plant cells. However, in some circumstances water flow might limit growth rate, particularly in tissues where the source of water (e.g. xylem vessels) is far removed from the cells controlling growth (e.g. epidermal cells). In droughted plants, too, water flow might limit expansion of new cells.

An equation encompassing this interplay between entry of water and wall relaxation would therefore be useful for analysing growth experiments. To achieve this, Equations 4.10 and 4.11 can be combined to form an equation describing water uptake and cell wall relaxation in a single cell in solution as follows:

Although rather complex, this equation marries the dual require-ments for water acquisition and irreversible wall expansion in growing cells. Accurate estimates for the variables can be made in single cells with the aid of a cell pressure probe. Heterogeneity within multicellular tissues reduces the accuracy of measurements, especially of conductivity. Equation 4.12 reinforces the point that any analysis of growth must focus on whether *L* exceeds φ substantially enough to reduce the right side of Equation 4.12 to Equation 4.11.

By postulating that Equation 4.11 provides a model for cell expansion, an appraisal can be made of the relative importance of cell wall properties (φ and *P*_{th}) and turgor (*P*) in determining growth rates. Equation 4.11 can be expressed as a linear relationship between d*V*/d*t* and *P*. However, viewing it as an equation for a straight line suggests that the parameters φ and *P*_{th} are rather less variable than we now believe them to be; the Lockhart equation is better viewed as a model linking three variables (φ, *P*_{th} and *P*) which can modulate growth in response to developmental or environmental cues.