1. In this paper, a theoretical framework for the analysis of growth is described. Growth is equated with change in volume (V) and the growth rate is given by the equation; dV/dt = (dm/dt)(1/ρ) − (dρ/dt)(m/ρ2) where m is the mass and ρ the density. The volume is inclusive of internal air spaces.
2. The second term of the growth equation (see above) can be ignored if density is constant over time. Data for humans (and presumably other large animals) show that while composition changes over time, the density is approximately constant at about that of water. In that case, the growth rate can be estimated from measures of the rate of change of mass. However, the density of plants is variable (c. 0·4–1·2 g cm−3) and measures of mass and density are necessary to analyse plant growth.
3. To use the theory as the basis of plant growth models, it is necessary to develop simple methods for estimating the surface area of roots, stems and leaves assuming that the mass and volume are known. A literature review found that the surface area to volume ratios of leaves and roots generally increase with the mass concentration of water. Theoretical arguments are used to predict that in woody stems, the situation should be reversed such that the surface area to volume ratio increases with the mass concentration of dry matter. Those relationships should be very useful in the development of plant growth models.
4. Measures of plant dry mass and estimates of the rate of change in dry mass are shown to be very difficult to interpret because of differences in the mass concentration of dry matter between individuals and over time.
5. It is concluded that measures of mass and density will be necessary before plant growth analysis can achieve its full potential. A framework for extending the theory to include the forces necessary for growth to occur is described.
Understanding growth, whether from a biochemical, biophysical, mechanical or morphological perspective, is a central theme of the biological sciences. Despite that, there is no generally accepted method for measuring and hence analysing growth.
At the cellular scale, growth is normally measured on a volumetric basis, although examples exist where the growth of unicellular organisms has been expressed on both a mass and dry mass basis. Plant scientists usually measure the growth of herbaceous and/or small woody plants in terms of dry mass but measurements of volume are used for larger plants such as trees. Animal scientists usually assess growth using measures of mass. In each case there are obvious practical reasons for adopting a particular measure. For example, volume is usually more difficult to measure than mass. In addition, the mass of small plants can be difficult to measure accurately because of changes in the water content and for trees, volume is a practical measure as it is non-destructive. In the case of animals, mass is an obvious choice because measurement of the dry mass poses ethical and logistical problems. While the above-noted practical constraints are important, it is also important to have a clear theoretical basis for measuring and hence analysing growth.
Here, a theoretical basis for the measurement of growth is described. That theory is then applied to the modelling and analysis of plant growth.
Some preliminary considerations
Newton's approximation (rather than Einstein's more accurate description) that space, time and the quantity of matter are independent of each other is adopted as the basis for the description of growth. A consequence of the Newtonian framework is that matter is viewed as moving through space, rather than space through matter, and volume is established as the primary measure of growth.
Having adopted volume as the appropriate state variable, it is necessary to identify the volume of an organism. That is (theoretically at least) simple for single cells because the volume is defined by the outer edge of the cell. In the case of multicellular organisms, the presence of internal air spaces complicates the identification of an organism's volume. Many internal air spaces have obvious physiological roles, e.g. gas exchange often occurs in humid internal spaces, such as lungs or the internal air spaces in leaves. Buoyancy control in marine organisms (e.g. swim bladders in fish) is another well-known example of the role played by internal air space. However, internal air spaces may also be present for other reasons. For example, geometric and mechanical factors seem to be the primary reason for the presence of intercellular spaces at the junction between cells (Thompson 1942; Jarvis 1998). Despite the different roles and/or reasons for the presence of internal air spaces (Raven 1996), they are included in the volume as defined here, because the outer edge effectively defines the amount of habitat occupied by an organism. The effects of other small organisms (e.g. bacteria) within that space are not considered and lie beyond the scope of this paper.
From a conceptual viewpoint it is useful to segment organisms into three spatial components, air space (a), solution (u) and structure (s). That approach, called the a–u–s–V scheme was originally developed to describe leaves (Roderick et al. 1999a) but it is general enough to be applied to other organs or organisms (e.g. lungs, vascular tissue, skeleton, etc.). In that scheme, the volume (V, m3) of an organism is:
The a–u–s–V scheme is useful because it explicitly recognizes the presence of internal air spaces. It also separates the solution, which is the major source of metabolic activity, from the structure whose role is primarily mechanical.
The growth equations
It follows from the above discussion that the volume (V) of an organism which has grown from an initial size (V0) over a time interval (t) is
where m (kg) is the mass, ρ (kg m−3) the density and d(m/ρ)/dt (m3 s−1), which is equivalent to dV/dt, is the growth rate. To the first order, the growth rate is:
Describing the components
Density is the ratio of the total mass within a given space to the volume of that space. Confusion sometimes arises when a measurement is described as the density but is actually the mass concentration of some component. For an organism containing k components, the density and mass concentration are related by:
In some cases, such as chemical analyses, the mass fraction of various components may be measured. The mass fraction (f) of a given component is defined as
and is unitless. The mass concentration and mass fraction of a component are related by:
For comparisons over time and/or between different individuals, the mass concentration is more useful than the mass fraction because it accounts for differences in density.
Distinguishing between growth and form
The theoretical framework described above treats organisms as ‘blobs’ of known volume and mass but of unknown form. However, form, which can be described by the shape and dimensions (Niklas 1994), is critical to the way in which an organism interacts with the environment (Thompson 1942). In growth analysis, it is important to distinguish between measures of growth and the linear or areal dimensions of form. For example, the height of either a tree or a human is a function of volume and form; change in height is not a measure of growth. One way to distinguish between measures of growth and the dimensions of form is via the units, i.e. measures of growth will have volumetric (m3) units, while the dimensions of form will have linear (m) or areal (m2) units. As a simple example, consider a cylinder of height h and radius r which is growing over time. The volume is given by
and the growth rate is
For terms on the right in equation 8, the dimensions are [L][L][L][T]−1 and the units of growth rate are m3 s−1.
Nevertheless, growth and form are closely linked as summarized in the following statement, where in one instance the original word growth has been replaced by the word in brackets.
‘In short it is obvious that the form of an organism is determined by its rate of (expansion) in various directions; hence rate of growth deserves to be studied as a necessary preliminary to the theoretical study of form, and organic form itself is found, mathematically speaking, to be a function of time.’ (Thompson 1942; p. 79).
The growth equations (equations 2 and 3) are obvious and hence simple. However, as far as I am aware, those equations are not used for growth analysis although the basic framework has been recognized previously. For example, in the plant sciences, Levitt (1969) equated growth with volume change and noted that mass change would be a satisfactory estimate of growth only if density remained constant over time. Thompson (1942) assumed that the density of an individual would remain approximately constant over time. Where that is an appropriate assumption, the second term in equation 3 can be ignored and the growth rate will be proportional to the rate of change of mass and inversely proportional to density. The validity of that assumption is investigated in the following section.
Variations in density and composition
The casual observation that many terrestrial and marine animals, regardless of age, will more or less float in water implies that their density must be reasonably close to that of water. Measurements of humans are consistent with those observations (Table 1). In humans, variations in density between individuals are largely owing to variations in the volumetric fractions of fat (ρ ≈ 0·9 g cm−3), lean tissue such as muscle (ρ ≈ 1·1 g cm−3) and internal air space (Malina & Bouchard 1991). Note that it is difficult to use casual observations to estimate the density of very small aquatic organisms because their motion is dominated by viscous forces because of their large surface area to volume ratio (McMahon & Bonner 1983).
Table 1. Estimates of the density and composition of human males (from Malina & Bouchard 1991). The estimates of density apparently exclude the air space which is at odds with the definition adopted in this paper. If the internal air space was included, the density estimates shown above would decline slightly. According to Malina & Bouchard (1991), fractional air space within humans varies with age, sex and stature
Infant at birth
Density (g cm−3)
Mass fraction of water
Mass concentration of water (g cm−3)
Whilst the density of humans (and presumably many other animals) is relatively conservative, the composition of the human body changes over time. In particular, decreases in the mass concentration of water with age (Table 1) are usually accompanied by increases in the mass concentration of protein, fat and minerals (Malina & Bouchard 1991). Many other changes also occur, e.g. changes in bone density with age and nutrition (Rodahl, Nicholson & Brown 1960).
Estimates of the range in density and the mass concentration of dry matter for terrestrial plants are listed in Table 2. The estimates of leaf density probably provide a realistic range for the density of stems and roots. Note that if the mass (but not the volume) of the internal air space is ignored, then
Table 2. Estimates of the range in density (ρ) and mass concentration of dry matter ([D]) in various plant compartments. With the exception of woody stems, these estimates are compiled from limited data sources and are subject to future revision
where [D] and [Q] are the mass concentration (kg m−3) of dry matter and liquid, respectively. In terrestrial plants, the mass fraction of liquid (Q) is usually measured instead of [Q]. Q can vary diurnally, seasonally and with environmental conditions (Slatyer 1967) but it usually declines as a plant grows (Evans 1972; Fig. 24.12). Within leaves, much of the variation in Q appears to be the result of differences between species (Hughes, Cockshull & Heath 1970). There is evidence from one study (Sims, Seemann & Luo 1998; see Discussion in Roderick et al. 1999b) that the density of leaves from a given species may be relatively constant but further measurements are necessary to establish whether that is true. The density of cell-wall material from woody stems is known to be almost constant, irrespective of species at c. 1·5 g cm−3 (Desch 1973). [Dstem] is usually, but not always (Zobel & van Buijenen 1989), relatively constant (c. ± 0·15 g cm−3) for a given species where the within-species variation is the result of a combination of genetic and environmental factors (Zobel & van Buijenen 1989; Lindstrom 1996). Within an individual tree, [Dstem] usually increases with radial distance from the centre of the stem but decreases with height (Zobel & van Buijenen 1989). Consequently, [Dstem] usually increases with age (Zobel & van Buijenen 1989).
In summary, the density of humans and presumably many other animals appears to be relatively constant at about the density of water. Thus, estimates of the rate of change of mass can be used to estimate the growth rate and those estimates should be more or less comparable between different individuals. Within humans, [Q] usually declines over time and that change in composition is presumably correlated with changes in human growth rates. A decline in [Q] with time is also a common feature of plant growth and should also be correlated with changes in growth rates. However, there are large variations in the density of whole plants and their components, and those variations will lead to relatively large differences in growth rates among individuals.
Application to the modelling and analysis of plant growth
The general principles of vascular plant function, i.e. specialized components (leaves, stems, roots, reproductive structures) which are linked by a vascular system, are well known. A starting point for modelling plant growth based on those principles is to express the plant volume as
where each component could in turn be separated into volumetric fractions of air space, solution and structure as required. There are many mechanical, physiological and biological factors which must be considered when seeking general solutions to equation 10 (Niklas 1992, 1994, 1997). From a modelling perspective, an obvious problem is to estimate the surface area of leaves and roots assuming that their volume and mass are known.
Composition and morphology of plant components
For typical leaves from herbaceous and woody C3 species it has been found that (Roderick et al. 1999b): (1) the maximum photosynthetic rate per unit (leaf) volume is positively correlated with, and very nearly proportional to, the surface area to volume ratio; (2) the surface area to volume ratio is positively correlated with the mass fraction of liquid; (3) the mass of leaf nitrogen (N) is a relatively constant fraction of the liquid mass; (4) the mass of leaf carbon (C) is a relatively constant fraction of the mass of dry matter.
Consequently, most of the properties which are usually considered functionally important in leaves are correlated with the mass fraction of liquid and the liquids are mostly water.
The rate of uptake of water and nutrients from the soil is proportional to the root surface area (Larcher 1995). According to Ryser (1996), as grass roots get thinner (i.e. surface area to volume ratio increases), the mass concentration of dry matter declines. Assuming the density of those roots was relatively conservative at around 1 g cm−3, it follows that the surface area to volume ratio must increase with the mass concentration of liquid. Because most roots approximate tapered cylinders, it also follows that the root length per unit root volume must increase with the mass concentration of liquid (see data in Ryser 1998).
The basic function of tree stems is to provide mechanical support and a transport network linking roots with leaves. [Dstem] is the best predictor of wood strength (Desch 1973; Zobel & van Buijenen 1989). Mechanical models predict that as [Dstem] increases, the stem could be potentially taller for any given diameter at the base (Schniewind 1962). However, because the density of cell wall material is approximately constant, it follows that as [Dstem] increases, the free space per unit stem volume in which other substances (e.g. water) can fit must decline (Siau 1984; Skaar 1988). Consequently, [Dstem] should capture some of the trade-off between the mechanical and hydraulic functions of tree stems. That proposition is consistent with the negative correlation between permeability and [Dstem] (Booker & Kininmonth 1978).
In summary, the morphology and composition of various plant components are correlated. The mass concentration of liquid generally increases with the surface area to volume ratio in typical leaves and roots. Within woody stems this relationship should be reversed, so that the surface area to volume ratio increases with the mass concentration of dry matter. (Note that in contrast with leaves and roots, the surface area to volume ratio of plant stems has no direct functional significance.) Those relationships highlight the importance of water in linking the composition and morphology of separate components with their function. The importance of water is not surprising, because it is the major component of most cells and the motion of water also links the separate components of plants via the vascular system.
Interpreting analyses of changes in dry matter
It is usual practice in many plant science disciplines to measure and subsequently analyse experimental results in terms of the rate of change of plant dry mass. It would be useful if a procedure was developed to estimate the growth rate (i.e. dV/dt) from those data and that is attempted in this section. First, some terminology is necessary.
The mass of a plant (m) can be expressed as:
where the subscripts denote the mass of dry matter (d) and liquid (q), respectively, and the mass of the air has been ignored. The mass concentration of dry matter ([D]) and liquid ([Q]) are
where the volume (V) includes the air space. The mass fractions of dry matter (D) and liquid (Q) are;
It is existing practice to estimate a quantity which is best described as the relative rate of change of dry mass (R′) and defined as;
R′ can be related to the growth rate by differentiating equation 12 as follows,
The rate of change of volume per unit volume is the relative growth rate (R). Thus,
To estimate R from published estimates of R′ it is necessary to estimate the magnitude of the second term in equation 16b. [D] is always positive but varies by an order of magnitude between different plant species (Table 2). Whilst [D] may increase or decrease in the short term (e.g. drought, rainfall), it would usually increase over the longer term. Without measurements of [D] there is no simple way to estimate R from estimates of R′.
The above analysis raises the question: how should estimates of R′ be interpreted? The principal difficultly in interpreting R′ is that many growth processes (e.g. cell division, cell expansion, metabolic activity) occur in solution. However, liquids are excluded from the growth analysis. Here, a combined numerical–physiological approach is used to understand how R′ should vary.
From eqn 14, it follows that over any given time interval, a faster R′ will result from either a larger dmd, smaller md or both. Most of the dry mass in plants is ultimately made from the products of photosynthesis. Assuming that respiration is a relatively conservative fraction of total photosynthesis (Landsberg & Gower 1997; Reich et al. 1998), it follows that dmd/dt should generally increase with photosynthetic rates and the maximum photosynthetic rates per unit leaf volume increase with the surface area to volume ratio which in turn increases with Q (Roderick et al. 1999b). If that relationship also held at the canopy scale, then the CO2 assimilation rate (and hence dmd/dt) should generally increase with the mass of water in the canopy. I am unaware of any direct tests of that relationship, although transpiration is approximately proportional to the total green mass of the canopy (Larcher 1995; Fig. 4.50, p. 268). Schulze et al. (1994) found that the rates of CO2 assimilation and water vapour loss were proportional to the mass of nitrogen per unit dry mass (Nd) in the leaves. Because Nd increases with Q (Fig. 1), it follows that CO2 uptake (and water vapour loss), and hence dmd/dt, should approximately increase with the total mass of liquid in the canopy. In terms of the denominator, md will be minimized if D is small, from which it follows that Q would be large. Consequently, plants with the fastest R′ should in general have the highest Q (also see calculations in Appendix). That prediction is consistent with observations from several studies (Garnier 1992; Ryser & Lambers 1995; Ryser 1996; Ryser & Aeschlimann 1999).
Many studies have shown that R′ usually increases with specific leaf area (SLA) which is the ratio of projected leaf area to leaf dry mass. However, SLA increases with Q (Hughes et al. 1970;Stewart et al. 1990; also see Fig. 1). Similarly, many studies have shown that R′ increases with Nd. However, as noted above, Nd increases with Q. Thus the previous interpretation that R′ should generally increase with Q is consistent with conventional interpretations based on SLA and Nd, because both of those quantities increase with Q.
Note that water is not stiff (i.e. viscous) enough to support large structures (Niklas 1992). Consequently, [Q] should generally decrease as V increases and the growth rate of a plant should also depend on V for mechanical reasons.
An extraordinary feature of the growth of humans (and presumably other animals) is that while the composition changes over time, the density remains approximately constant at about that of water. Consequently, the growth rate (dV/dt) can be estimated using measures of the rate of change of mass. However, that is not the case for plants. While the composition of plants also changes in much the same way over time (i.e. decrease in Q), and the density of an individual plant may be constant over time, there are relatively large differences in density (c. 0·4–1·2 g cm−3) among plants. Consequently, measures of mass and density are both necessary to analyse plant growth.
Making accurate estimates of the mass and density of small plants is difficult because of the relatively rapid changes in liquid content which can occur. However, just because mass and density are difficult to measure, does not mean that they are the wrong things to measure. Often, it may be that the measurement errors are quite large, particularly for measurements of roots (e.g. Habib & Chadoeuf 1989). Nevertheless, the theory shows that measurements of mass and density are essential if plant growth analyses are to be interpreted correctly. Given the complexities involved, a practical way forward would be to systematically assess whether the density of leaves, stems and roots is relatively constant for a given species. If that was the case, then the number of essential measurements could be reduced.
As noted in the Introduction, foresters measure and analyse tree growth on a volumetric basis. It is generally found that fast-growing trees usually produce wood with a small mass concentration of dry matter ([Dstem]). An alternative way of expressing that relationship is that trees with a small [Dstem] are usually fast growing. That relationship can be expressed mathematically by ignoring the second term in equation 3 and combining that with equation 9 to give:
Assuming that the approximation in equation 17 holds, it follows that an inverse relationship between the growth rate of a stem (dVstem/dt) and [Dstem] is almost true by definition. However, some studies have disputed an inverse relation between growth rate and [Dstem]. Zobel & van Buijenen (1989) have reviewed this topic and noted that most studies investigating the relation between growth rate and [Dstem] have used changes in the stem radius as the measure of growth. That approach is invalid because the rate of change of stem radius (i.e. dr/dt in equation 8) is not the growth rate. [Dstem] has to be correlated with dVstem/dt if a relationship is sought with the growth rate of the stem. That example highlights the importance of separating measures of morphology from measures of growth.
There are still many basic scientific questions to be addressed about growth. For example, it has long been known that the growth rate of trees declines with age, while wood scientists have long known that [Dstem] increases with age (Zobel & van Buijenen 1989). Increases in [Dstem] must eventually lead to increases in the resistance to flow through the stem as a result of a reduction in available space per unit stem volume. Increased resistance to flow through the stem does limit the photosynthetic uptake per tree in old trees (Yoder et al. 1994; Ryan & Yoder 1997; Hubbard, Bond & Ryan 1999). Those results are consistent and pose the following questions: (1) can the decline in tree growth rate with age be attributed to an increase in the volumetric fraction of structure within the tree; (2) is the increase in the volumetric fraction of structure necessary for mechanical support? These are just some of the many questions which can be addressed using appropriate measurements. Perhaps when those measurements become available, we may find relatively simple explanations for many growth-related phenomena that at the moment appear to require complicated explanations.
The theoretical framework outlined in this paper describes the changes in mass and volume as an organism grows but the forces which are necessary for work to be carried out in expanding the organism have not been considered. Thermodynamics could be used to link the growth equations with the forces. That should be relatively simple in the plant sciences because there is a large literature linking differences in the chemical potential of water with various aspects of growth. However, a complete thermodynamically based theory of growth will also need to include the internal air space. That space contributes little mass to an organism. However, internal air spaces are important for gas exchange and they can change both slowly (e.g. Malina & Bouchard 1991) and quickly (e.g. Jackson & Armstrong 1999) over time. They can also make up a large fraction of the volume occupied by an organism. Perhaps most important, changes in internal air space are a simple means to alter simultaneously the relationship between the internal and external surface area, mass and volume. Internal air spaces should be very important for the reasons noted above.
Comments by Professors Ian Noble and Ralph Slatyer on an earlier and somewhat different version of this manuscript are appreciated. Further comments by two anonymous reviewers improved the manuscript. I am indebted to colleagues, Sandra Berry and Dr Ashley Sparrow for many stimulating discussions on the nature of growth.
Appendix. Comparison of relative growth rate (R) and relative rate of change of dry mass (R′)
Interpreting differences in R′ between plants is very difficult because the computation is confounded by differences in the mass concentration of dry matter between plants. This is highlighted using the hypothetical example shown in Table 1, where the growth rate (dV/dt), the relative growth rate [R = (dV/dt) (1/V)] and the rate of change of dry mass (dmd/dt) are the same for the two plants. However, the relative rate of change of dry mass [R′ = dmd/dt)(1/md)] is three times faster for plant 2 than plant 1. That is a direct consequence of plant 2 having a small mass concentration of dry matter, which leads to a smaller value of md, and ultimately results in a larger value of R′. For that (mathematical) reason, plants with the least mass concentration of dry matter will generally always have the fastest R′ provided dmd is positive.
Table A1. Calculation of the absolute and relative rates of change using hypothetical measurements (made over three equal time intervals) of two plants having the same volume and density but a different mass concentration of dry matter. Absolute (dV/dt, dmd/dt) and relative rates of change (R, R′) are instantaneous estimates at t2 (time 2)