# Topic 4.3 Calculating Velocities of Water Movement in the Xylem and in Living Cells

Let's begin by noting that the velocity with which water travels up the trunk of a tree depends on both the type of tree and the transpirational demand placed on the xylem. For trees with wide vessels (radii of 100 to 200 µm), peak velocities of 16 to 45 m h^{–1} (4 to 13 mm s^{–1}) have been measured. Trees with smaller vessels (radii of 25 to 75 µm) have lower peak velocities, from 1 to 6 m h^{–1} (0.3 to 1.7 mm s^{–1}). For our calculation we will use a figure of 4 mm s^{–1} for the xylem transport velocity and 40 µm as the vessel radius. This is a high velocity for such a narrow vessel, so it will tend to exaggerate the pressure gradient required to support water flow in the xylem.

A version of Poiseuille’s equation (see textbook Equation 4.2) can be used to estimate the pressure gradient (Δ*Ψ*_{p}/Δ*x*) needed to move water at this velocity (4 × 10^{–3} m s^{–1}) through a pipe of radius (*r*) 40 µm. By dividing textbook Equation 4.2 by the cross-sectional area (π*r ^{2}* ) of the xylem vessel, we find that the rate of transport (

*J*

_{v}, in m s

^{–1}) is given by the following equation:

Taking the viscosity of xylem sap to be that of water (10^{–3} Pa s), we find that the pressure gradient required is 2 × 10^{4} Pa m^{–1} (or 0.02 MPa m^{–1}). This is the pressure gradient needed to overcome the viscous drag that arises as water moves through an *ideal* vessel at a rate of 4 mm s^{–1}. *Real* vessels have irregular inner wall surfaces and constrictions, such as perforati on plates, at the points where vessel elements meet. Tracheids, with their smaller diameters and pitted walls, offer even greater resistance to water flow. Such deviations from an ideal pipe will increase the frictional drag above that calculated from Poiseuille's equation, but since we selected a low value for vessel radius, our estimate of 0.02 MPa m^{–1} should be in the correct range for pressure gradients found in real trees.

Let's now compare this value (0.02 MPa m^{–1}) with the driving force that would be necessary to move water at the same velocity through a layer of *living cells.* We will ignore water movement in the apoplast pathway in this example and focus on water moving from cell to cell, crossing the plasma membrane each time. As we described in textbook Chapter 3, the velocity (*J*_{v}) of water flow across a membrane depends on the membrane hydraulic conductivity (*Lp*) and on the difference in water potential (Δ*Ψ*_{w}) across the membrane:

A high value for the *Lp* of higher plant cells is about 4 × 10^{–7} m s^{–1} MPa^{–1}. Thus, to move water across a membrane at 4 × 10^{–3} m s^{–1} would require a driving force (Δ*Ψ*_{w}) of 10^{4} MPa (4 × 10^{–3} m s^{–1} divided by 4 × 10^{–7} m s^{–1} MPa^{–1}). This is the driving force needed to move the water across a *single membrane*. To move through a cell, water must cross at least two membranes, so the total driving force across one cell would be 2 × 10^{4} MPa. If we estimate the cell length as 100 µm (10^{–4} m, a generous estimate), then the water potential gradient needed for water to move at a velocity of 4 mm s^{–1} through a layer of cells would be 2 × 10^{4} MPa divided by 10^{–4} m, or 2 × 10^{8} MPa m^{–1}. This is an enormous driving force, and it illustrates that water flow through the xylem is exceedingly more efficient than water flow across the membranes of cells. Comparing the two driving forces (for open vessel and cell transport), we see that the two pathways show a difference of a factor of 10^{10}. This is a huge difference indeed.