A Biophysical Analysis of Stem and Root Diameter Variations in Woody Plants

Geometry

The stem (root) diameter, D (m), equals the diameter of the outer cylinder in Figure 1. The mature xylem with a diameter D′ (m) is represented by the inner cylinder. The thickness of the storage compartment is equal to:ρ=(D−D′)/2 Equation 1The volume of the storage compartment (V in m³) depends on its thickness:V=πρD′l(1+ρ/D′) Equation 2where l is a constant equal to the length unit of the axis. A first approximation is carried out to simplify further analyses, where the thickness ρ is assumed to be much lower than the diameter D′. This makes it possible to transform equation 2 into:V≅πρD′l Equation 3Using an empirical relationship between ρ and D (see Eq. 13), it can be shown that the error resulting from this approximation is always less than 10% and it is less than 5% for diameters greater than 5 cm.

Diameter Variation Components

The variation in diameter D is the result of elastic and thermal expansion and growth processes:dD/dt=(dD/dt)el+(dD/dt)th+(dD/dt)gr Equation 4Assuming that elastic expansion does not affect the xylem diameter D′, the first term in Equation 4 is defined by Equations 1and 3 as:(dD/dt)el=2(dρ/dt)el=2(ρ/V)(dV/dt)el Equation 5The relative elastic variation of the storage compartment volume is proportional to the variation of the turgor pressure P (MPa) in this compartment (Dale and Sutcliffe, 1986):(dV/dt)el/V=(dP/dt)/ɛ Equation 6where ɛ (MPa) is the elastic modulus. The elastic modulus increases with turgor and cell size (Tyree and Jarvis, 1982; Dale and Sutcliffe, 1986) and reaches an asymptote for high turgor and cell size. For the sake of simplification and to limit the number of parameters, a linear relationship was assumed. To take both relationships into account, the elastic modulus was assumed to be proportional to turgor and diameter:ɛ=ɛ0PD Equation 7where ɛ₀ (m⁻¹) is a parameter.

Combining Equations 5, 6, and 7, the diameter variation resulting from elastic expansion was:(dD/dt)el=2ρ(dP/dt)/(ɛ0PD) Equation 8In physics, the thermal expansion of the material (including wood) is described as being proportional to temperature (T in K). We considered that this law could be applied to a living plant and that the relative diameter variation resulting from temperature fluctuations was proportional to temperature change:(dD/dt)th/D=α dT/dt Equation 9where α is the coefficient of thermal expansion (K⁻¹).

The effect of growth may be represented as:(dD/dt)gr=(dD/dρ)(dρ/dt)gr Equation 10Because D′ can be considered to be constant compared with ρ for plastic growth on an hourly basis, the differentiation of V in Equation 3 is:(dρ/dt)gr=ρ(dV/dt)gr/V Equation 11When the turgor pressure P exceeds a threshold value Y (MPa), irreversible plastic growth occurs, as described byLockhart's (1965) equation:(dV/dt)gr/V=φ(P−Y)if P>Y Equation 12 (dV/dt)gr/V=0 if P≤Y where φ (MPa⁻¹ sec⁻¹) is the extensibility of the cell walls.

Because the time frame of the model is longer than an hour, the growth of D′ has to be considered to calculate d D/ dρ (Eq. 10). To take the growth of D′ into consideration from a mechanistic point of view, it would be necessary to consider the process of differentiation of mature xylemic vessel, leading to a specific model which is not within our subject matter. That is why an empirical approach was used, based on the fact that the thickness of extensible tissues increases with the diameter of the organ, as shown for apple stems by Huguet (1985). An empirical relationship between ρ and D was used to calculate d D/ dρ:ρ=a(1−exp[−bD]) Equation 13where a and b are parameters. Thus, it can be deduced that:(dD/dt)gr=(dD′/dt)gr+2(dρ/dt)gr=(dρ/dt)gr/ Equation 14 [b(a−ρ)] Combining Equations 11, 12, and 14, the diameter variation resulting from growth was obtained as a function of turgor pressure and ρ:(dD/dt)gr=ρφ(P−Y)/[b(a−ρ)](if P>Y) Equation 15 (dD/dt)gr=0 (if P≤Y)

Thickness of the Storage Compartment

For the sake of simplicity, it is assumed that the flow of water to the storage compartment is essentially a flow of xylem water. The change of volume ([d V/ d t]_f) resulting from this flow is described by an equation derived from nonequilibrium thermodynamics (Katchalsky and Curran, 1965):(dV/dt)f=AL[Px−P−ς(πx−πs)] Equation 16where L (m MPa⁻¹ s⁻¹) is the radial hydraulic conductivity of the membrane separating the stem storage compartment from the xylem, A (m²) is the surface area of this membrane, P_x (MPa) is the hydrostatic pressure in the xylem, π_x (MPa) is the osmotic pressure in the xylem, π_s (MPa) is the osmotic pressure in the storage compartment, and ς is the reflection coefficient of the membrane to the solutes. Because π_x≪ π_s, it can be disregarded in Equation 16 and in the following calculations.

The total variation of the storage compartment thickness includes variations resulting from the horizontal flow of water and thermal expansion:dρ/dt=(dρ/dt)f+(dρ/dt)th Equation 17with (dρ/d t)_f = ( ρ/V)(d V/ d t)_f and (dρ/d t)_th = ρα d T/ d t (from Eqs. 5, 9, and 11), which leads to:dρ/dt=L(Px−P+ςπs)+ρα dT/dt Equation 18combining Equations 16 and 17. The variation of ρ was dependent on turgor pressure and osmotic potential that had to be calculated.

Turgor Pressure and Osmotic Potential

The variation of the turgor pressure results from the flow of water to the storage compartment. The balance between the solution inflow and elastic-plastic changes of the volume (d V/ d t)_f = (d V/ d t)_el + (d V/ d t)_gr can be expressed using Equations 8, 12, and 16:AL(Px−P+ςπs)=V[(dP/dt)/(ɛ0PD)+ Equation 19 φ(P−Y)]if P>Y AL(Px−P+ςπs)=V[(dP/dt)/(ɛ0PD)]if P≤Y The turgor pressure variation can then be calculated: dP/dt=ɛ0L(Px−P+ςπs)PD/ρ−ɛ0φPD(P−Y) Equation 20  if P>Y dP/dt=ɛ0L(Px−P+ςπs)PD/ρ if P≤Y According to the definition of Van't Hoff, the osmotic pressure is:πs=RTns/V Equation 21where R is the universal gas constant and n_s is the number of moles of solutes in volume (V). Differentiation of Equation 21 over a period of time gives:dπs/dt=(RT/V)(dns/dt)−πs(dV/dt)/V+πs(dT/dt)/T Equation 22Considering a cell as a closed system where the amount of solutes, n_s, does not change with time and where the temperature is constant, only the second term in the right-hand side of Equation 22 is usually taken into account (Dainty, 1976). In higher plants, the uptake of solutes has to be accounted for, as described by the first term in the right-hand side. Daily and seasonal temperature variations lead to the last term of Equation 22. Assuming that solutes are transported to the growing stem by means of an active (and/or facilitated) mechanism, and using the Michaelis-Menten equation to describe the rate of the process, we have: d n_s/d t =ZXVC _p/ (K_M + C_p), where C_p is the solute concentration in the sap, Z is the maximum rate of the solute accumulation process considered to be constant, and X is the proportion of these solutes that are not consumed by respiration catabolism and remain soluble. If as a first approximation, K_M ≪ C_p, d n_s/d t ≈ ZXV, and considering Equation 16 for volume variations, Equation 22 can be rewritten as:dπs/dt=γT−πsL(Px−P+ςπs)/ρ+πs(dT/dt)/T Equation 23where γ = RXZ (MPa s⁻¹K⁻¹) is a parameter.

Governing Equations

By combining Equations 4, 8, 9, 15, and 20, we obtain Equation24:dD/dt=2L(Px−P+ςπs)+Dα(dT/dt)+ρφ(P−Y){1/[b(a−ρ)]−2} Equation 24Due to the restrictions in Equation 15, the last term in Equation24 is equal to 0 if P ≤ Y.

Equations 18, 20, 23, and 24 form a system of differential equations for P, π_s, ρ, and D, which can be solved numerically with given (inputted) functions of time P_x(t) and T(t), and the initial values of respective variables.

Initial Conditions and Parameterization

To run the model, initial values were needed for P, π_s, and ρ. These values were approximated at the end of the night before the beginning of the simulation. At this time of the day, we can assume that the water potential of the storage compartment was equal to the xylem hydrostatic pressure (P_x). In this case, the initial value of π_s can be calculated as:πs(0)=P(0)−Px(0) Equation 25At this time of the day, the osmoregulation is probably low because the water potential is high and fairly stable. Under these conditions, the turgor pressure can be proportional to the hydrostatic pressure of the xylem as shown by Fanjul and Rosher (1984) on apple leaves:P(0)=β[Px(0)−Px(P(0)=0)] Equation 26where β is an empirical parameter estimated using the Generalized Reduced Gradient method in the calibration procedure and P_x(P(0) = 0) is the hydrostatic pressure of the xylem for which the zero turgor was reached. For P_x(P(0) = 0), we used the value given by Fanjul and Rosher (1984) for apple leaves (−2.9 MPa) under well-watered conditions.

We measured the initial value of D and used it to compute initial ρ by means of Equation 13. By measuring the thickness of extensible tissues on three peach cultivars (see “Materials and Methods”), we could estimate the parameters of Equation 13 through a nonlinear regression procedure, a = 2.968 10⁻³ m and b = 32 m⁻¹. The three cultivars followed the same general curve.

The wall-yielding threshold pressure, Y, has been observed in a variety of plant tissues (Green et al., 1971;Green and Cummins, 1974; Bradford and Hsiao, 1982) with values ranging from 0.1 to 0.9 MPa. Assuming that the threshold pressure had to be higher for stem or root tissues than for young tissues or individual cells on which most of the measurements had been done, we chose Y = 0.9 MPa.

The virtual composite membrane is composed of several cell layers leading to possible cell-to-cell and apoplastic water flow. The reflection coefficient of the apoplast is usually close to 0, whereas along the cell-to-cell path, the presence of the membrane leads to a reflection coefficient close to 1 (Steudle, 2000). The overall tissue reflection will then be between 0 and 1. In our single cell model, most of the water has to cross the cell membrane, which is why we have adopted a high reflection coefficient (ς = 1). Nevertheless, we evaluate its size through the sensitivity analysis.

The other parameters of the model (α, L, γ, ɛ₀, and φ) were estimated through the calibration procedure, using the Generalized Reduced Gradient method.

Calibration of Model

The parameter values and standard deviations are summarized in Table II. The coefficient of thermal expansion for peach stems was estimated on two stems, the diameter variations of which were measured before bud break. The evolution in time of diameter showed a decreasing trend of 2.61 10⁻⁸ and 6.17 10⁻⁸ m s⁻¹, depending on the stem. Considering this decrease, α (Eq. 9) was estimated to be equal to 9.39 10⁻⁵ K⁻¹, which is close to the coefficients of thermal expansion for wood across the fibers (3–7 10⁻⁵K⁻¹) given by Forsythe (1954) andKoshkin and Shirkevitch (1975). The overall percentage variation explained by the optimized curve fitting was 65%. Diameter variations due to temperature conditions and predicted by the model were lower than those measured (Fig. 2), probably because the temperatures recorded by the meteorological station had underestimated the maximal field temperatures. Measurement of surface stem temperature made with thermocouple at the same period of the year on other peach trees gave maximal temperatures 2.2°C to 4.8°C greater than the maximal temperature recorded by the meteorological station.

• Download figure
• Open in new tab
• Download powerpoint

Fig. 2.

Diurnal variations of stem diameter with temperature in two peach stems in February before bud break. Relative diameter is calculated as the ratio “diameter:diameter at the beginning of the day.” The thin lines represent the measurements and the thick lines represent the simulations.

The Gotheron severe water stress treatment (GSS) was used to estimate β (Eq. 26), radial hydraulic conductivity L (Eq.16), and elastic modulus parameter ɛ₀ (Eq. 7) for the stem. The diameter variations of three stems were measured over a period of 24 h. The diameter growth of the stems was low and the diameter variation resulting from growth was not considered, which means that γ and φ were set at 0. We estimated the empirical parameter β to be equal to 0.463. The parameter ɛ₀ was estimated to be equal to 1.026 10³m⁻¹, resulting in values for the elastic modulus ɛ ranging from 2 to 50 MPa. These estimates are within the range of the values obtained for giant algal cells (10–60 MPa) and higher plants tissues (0–30 MPa) as reported by Dainty (1976),Tyree and Jarvis (1982), and Dale and Sutcliffe (1986). Radial hydraulic conductivity (L) was estimated to be equal to 2.86 10⁻⁸ m MPa⁻¹s⁻¹, which is in the range of the values obtained for giant algal cells (1.86 10⁻⁸ − 2.78 10⁻⁴ m MPa⁻¹ s⁻¹) and from cells of higher plant tissues (1.0 10⁻¹⁰ − 1.67 10⁻⁴ m MPa⁻¹ s⁻¹) as reported byDainty (1976) and Dale and Sutcliffe (1986).

The overall percentage variation of stem diameter explained by the optimized curve fitting was 93% (Fig.3).

• Download figure
• Open in new tab
• Download powerpoint

Fig. 3.

Mean diurnal variations of peach stem diameter on July 9 for the GSS treatment used for model calibration on stems. The mean was calculated on three stems. Relative diameter is calculated as the ratio “diameter:diameter at the beginning of the experiment.” The thin lines are the measurements and the thick lines are the model simulations.

The root data set collected from September 2 through 16 was used to estimate the growth parameters (φ and γ in equations 14 and 23, respectively) because growth was well marked during this period. The parameters α, β, and L estimated for the stem were assumed to be acceptable for the root and ɛ₀ only had to be reestimated. Tissue elasticity was higher in the root (ɛ₀ = 0.363 10³ m⁻¹) than in the stem. The growth parameters were estimated to be γ = 9.25 10⁻¹⁰ MPa s⁻¹ K⁻¹ and φ = 3.19 10⁻⁷ MPa⁻¹ s⁻¹. The cell wall extensibility usually ranges from 8.33 10⁻⁶ to 5.56 10⁻⁵ MPa⁻¹ s⁻¹ (Hsiao et al., 1998), which is one order of magnitude higher than our estimate for the root tissues of plum. This is probably true because φ is usually measured on young and very extensible tissues, whereas we were working on 5-year-old roots.

The overall percentage variation of root diameter explained by the optimized curve fitting was 97% (Fig. 4).

• Download figure
• Open in new tab
• Download powerpoint

Fig. 4.

Variations of plum root diameter from September 2 through 16 used for model calibration on roots. Relative diameter is calculated as the ratio “diameter:diameter at the beginning of the experiment.” The thin lines represent the measurements and the thick lines represent the model simulations.

Model Test

To test the model, we used the data sets that had not been used for the calibration. The simulations of stem diameter variation in the Avignon and Gotheron experiments fit the observations quite well (Fig.5). The model was able to reproduce the effect of water stress treatments on stem diameter variations at the two sites. For the root, the test was done on July 30 through 31 and August 19 through 21 data sets. Given the fact that growth rate was different between these two periods, γ had to be reestimated for each period (γ = 9.83 10⁻⁹ MPa s⁻¹K⁻¹ and 1.01 10⁻⁸ MPa s⁻¹K⁻¹, respectively). The model also made it possible to reproduce the diameter variation with time (Fig.6) on an hourly and daily scale.

• Download figure
• Open in new tab
• Download powerpoint

Fig. 5.

Diurnal variations of peach stem diameter for the treatments used for the model test (Gotheron well irrigated [GI], Gotheron mid-water stress [GmS], Avignon well irrigated [AI], and Avignon stressed [AS]). Relative diameter is calculated as the ratio “diameter:diameter at the beginning of the day.” The thin lines represent the measurements and the thick lines represent the model simulations.

• Download figure
• Open in new tab
• Download powerpoint

Fig. 6.

Diurnal variations of plum root diameter for the two periods used for the model test. Relative diameter is calculated as the ratio “diameter:diameter at the beginning of the experiment.” The thin lines represent the measurements and the thick lines represent the model simulations.

Effect of Xylem Water Potential Variations on Storage Variables

As illustrated by the simulations performed on the plum root system for the period September 2 through 16, the daily variations of the xylem water potential resulted in lower daily variations of the turgor pressure and in much lower osmotic potential variations (Fig.7). The coefficient of variation computed for the whole period was equal to 64%, 25%, and 5%, for the xylem water potential, the turgor pressure, and the osmotic potential, respectively. The elastic modulus was also sensitive to the water potential variations (Fig. 7) with a coefficient of variation (25.5%) very close to that of the turgor pressure to which it was highly correlated (R = 0.99). The variation of osmotic potential, turgor pressure, and elastic modulus with the xylem water potential followed diurnal hysteresis loops (Fig.8). Nevertheless, these variables were highly correlated. The osmotic and water potentials were negatively correlated (R = −0.78) when the turgor and the elastic modulus were positively correlated with the xylem water potential (R = 0.96 and 0.95, respectively). Although a marked hysteresis was observed, the water potential of the storage tissues was highly correlated to that of the xylem (R = 0.9; Fig. 8).

• Download figure
• Open in new tab
• Download powerpoint

Fig. 7.

Variations of simulated turgor pressure (dotted line) and osmotic potential (unbroken line) of the storage compartment, and measured xylem water potential (broken line) from September 2 through 16 for plum root (A). The variations of the simulated elastic modulus are drawn in B.

• Download figure
• Open in new tab
• Download powerpoint

Fig. 8.

Relationships between simulated water potential, osmotic potential, pressure turgor, elastic modulus of the storage compartment, and measured xylem water potential. The simulations were performed from September 2 through 16 on plum root.

Sensitivity Analysis to Parameters

The sensitivity analysis was performed on the plum root system, using the environmental and water potential conditions of the period from September 2 through 16. The parameters of the model had a variation of ±20% and the effect of this variation on the mean daily diameter growth rate and the mean shrinkage were assessed (TableIII).

The model was very sensitive to the parameters β and P_x(P(0) = 0) used to estimate the initial value of P. A variation of these parameters induced an equivalent variation of the shrinkage and the growth rate variation was two or three times higher. This shows that a precise estimate of the initial P is needed to predict the actual daily growth rate and, to a lesser extent, the shrinkage. The shrinkage was not sensitive to the empirical parameters (a and b) involved in the relationship between ρ and D (Equation 13), contrary to what was observed for radial growth rate.

Shrinkage and growth rate were not sensitive to the variations of the coefficient of thermal expansion (α). Nevertheless, some effect on shrinkage was observed when α was decreased 10-fold.

The flow of water to the storage compartment depended on the hydraulic conductivity L, the variation of which (around 2.86 10⁻⁸ m MPa⁻¹ s⁻¹ ± 20%) had no effect on diameter growth rate and shrinkage. For very low conductivity values (below 1.39 10⁻⁸m MPa⁻¹ s⁻¹), daily growth rate and shrinkage decreased. It was more surprising that the daily growth rate also decreased for high conductivities (greater than 2.78 10⁻⁶ m MPa⁻¹ s⁻¹). According to the simulations, this can be explained by a turgor pressure which was often lower than the wall-yielding threshold pressure, Y. This low turgor pressure decreased the elastic modulus ɛ, and consequently increased shrinkage. Maximal growth was obtained for conductivity values ranging from 1.67 10⁻⁸ to 2.78 10⁻⁶ m MPa⁻¹ s⁻¹, which included the conductivity values (L = 2.86 ± 0.012 10⁻⁸ m MPa⁻¹ s⁻¹) estimated for peach stem. The 20% decrease of the reflection coefficient (ς) induced a 30% increase of shrinkage and a strong decrease of the growth rate (Table III). Shrinkage was not very sensitive to the variation of solute accumulation (parameter γ). Growth was positively but slightly dependent on γ variations. The elastic modulus variation had no effect on growth and a proportional effect on shrinkage. The shrinkage was not sensitive to the plastic growth parameters. Growth and cell wall extensibility varied likewise when change of the threshold yield Y had a stronger effect.

Hysteresis between Xylem Water Potential and Diameter Variations

The relationship between stem diameter and xylem water potential simulated by the model was similar to that obtained from experimental data as illustrated by the Avignon AI and AS treatments (Fig.9). Hysteresis was not marked on the well-irrigated treatment (AI) and no clear relationship between stem diameter and xylem water potential was observed. In case of water stress (AS), the general trend was an increase of stem diameter as xylem water potential increased. This general trend was disturbed by a strong hysteresis loop. For a given xylem water potential, the stem diameter is lower during a period of increasing water potential (between 8 and 14 h Greenwich Mean Time) than during a period of decreasing water potential, which is in agreement with the results obtained on plants as different as cotton (Klepper et al., 1971) and peach tree (Garnier and Berger, 1986). The hysteresis loop is caused by the delayed response of stem diameter compared with the xylem water potential. This delay results from the storage properties (growth, elasticity of tissues, and volume) and the radial hydraulic conductivity of the membrane separating the stem storage compartment from the xylem.

• Download figure
• Open in new tab
• Download powerpoint

Fig. 9.

Relationship between the relative stem diameter and xylem water potential for the Avignon experiment (AI and AS). Relative diameter is calculated as the ratio “diameter:diameter at the beginning of the day.” The thin lines represent the measurements and the thick lines represent the model simulations. Figures on the lines represent the time (Greenwich Mean Time) of the day.

To analyze the effect of these properties on the degree of hysteresis, the diameter variation was simulated using the root parameters on September 3, which was a typical day for that season. Hysteresis was equally pronounced whether growth was considered (φ = 0 and γ = 0) or not (Fig. 10). To analyze the effect of elasticity, volume of storage compartment, and conductivity, we multiply the values of ɛ₀, radial hydraulic conductivity (L) and initial thickness of the elastic tissues by 0.5 or 2. The effect was significant with a higher hysteresis value as ɛ₀ and L decreased (Fig.11). The initial thickness of elastic tissues resulted in higher variations of the hysteresis, which increased as thickness increased (Fig. 11).

• Download figure
• Open in new tab
• Download powerpoint

Fig. 10.

Simulated relationship between root diameter and xylem water potential for plum root on September 3, with (A) or without (B) growth.

• Download figure
• Open in new tab
• Download powerpoint

Fig. 11.

Sensitivity of hysteresis between root diameter and xylem water potential to elasticity parameter (ɛ₀), radial hydraulic conductivity (L), and initial thickness of elastic tissues (ρ₀) for plum root on September 3. Relative diameter is calculated as the ratio “diameter:diameter at the beginning of the day.” ɛ₀, L, and ρ₀ were multiplied by 0.5 or 2 as indicated at the top of the figure.