Introduction

Plants grow throughout most of their lives, and they use growth to acquire light, nutrients, and water. Organic compounds are metabolized and mineral elements are deposited in plant cells throughout the cellular development, maturation, and senescence. Net rates of biosynthesis and mineral deposition are needed to understand physiology and the response to environmental variation. Particularly in growing tissue it is necessary to consider expansion and displacement of cells relative to observed concentration patterns. This is because, in contrast to mature tissues, growing tissue has concentration profiles that are produced by cell expansion as well as nutrient import, catabolism and anabolism, thus, quantitative analyses of morphogenesis are required.

Plant computational morphodynamics is now a popular field of study. The classical literature on plant growth analysis is well cited. Landmark papers include an analysis of cell division rates based on a form of the continuity equation (Goodwin and Stepka 1945), the use of streak photographs by Erickson and Sax (1956) to quantify the one-dimensional growth rate pattern in corn roots, and a quantitative two-dimensional analysis of leaf growth (Erickson 1966). The important distinction between spatial and material (cell-specific) aspects of plant development was clarified three decades ago (Silk and Erickson 1978, 1979; Gandar 1983a, b). More recently Kwiatkowska and Dumais (2003) provided a quantitative analysis of apical morphogenesis, while improved computer-assisted tracking of natural marks has facilitated high-throughput growth analysis at high resolution (Walter et al. 2002; Grandjean et al. 2004; Reddy et al. 2004; Miller et al. 2007). At a recent workshop scientists showed time-lapse images from confocal microscopy of living tissue and described how image analysis is facilitated by genetic transformation to introduce fluorescent markers into plant cells (reviewed in Cunha et al. 2012, images viewable at http://www.computableplant.org/).

And yet in 2012, many computational models are time dependent but neglect to consider growth. Perhaps this is the heritage of the influential Turing model (1952) for morphogenesis. Turing showed that when two or three reactants with different diffusivities are involved in a common chain of chemical reactions, then a small amount of noise leads to stable, inhomogeneous patterns that are suggestive of biological patterning. This has been an appealing approach for theoretical biologists; by 2012 Turing’s paper had 3428 citations. However, the Turing models do not include expansion or active sinks both of which characterize growing plant organs. Another popular model, often used for mineral uptake in plants, is based on the Michaelis-Menten equations (1913) for enzyme kinetics. The Michaelis-Menten model assumes a well-stirred beaker: The system is seen to vary in time and to lack spatial gradients. This is not an appropriate pair of assumptions for a plant. The simplest model plant, in an appropriate (moving) reference frame, has spatial patterns of growth, anatomy, and biochemistry that may not change in time. The model plant must however have spatial variation associated with the developmental gradient, that is, the variation in tissue age at different positions.

By neglecting growth we are missing a major aspect of the ability of sessile organisms to respond to their environment. Continuity equations provide a way to couple growth velocity fields v with patterns of element concentration ρ in space and time, t, to improve mechanistic understanding of the interaction of growth processes and chemistry (Silk and Erickson 1979). In general the local deposition rate, d in a tissue can be calculated from

$$ d=\frac{{\partial \rho }}{{\partial t}}+\nabla \cdot \left( {\rho \boldsymbol{\upupsilon}} \right) $$
(1A)

which can be expanded to

$$ d=\frac{{\partial \rho }}{{\partial t}}+\rho \left( {\nabla \cdot \boldsymbol{\upupsilon}} \right)+\boldsymbol{\upupsilon} \cdot \nabla \rho $$
(1B)

When the deposition rate is positive, the location is a sink for the material of interest. The material can be imported or be produced by local metabolism. A negative deposition rate means the location is losing material by catabolism or by export either to the environment or to other plant tissues. The local deposition rate is a net accumulation rate, the result of synthesis, import, degradation, and/or export.

This article reviews deposition rates in plant tissue and some implications for physiology, ecology and molecular biology. The use of this approach to quantify cell division rates and the molecular regulation of cell division has been reviewed elsewhere (Fiorani and Beemster 2006); environmental effects on coordination between division and elongation have been reviewed by Tardieu et al. (2011). The article concludes by citing some recent computational advances that have the potential to greatly extend understanding of biochemistry and molecular biology of growing tissue.

Technical aspects

Non-growing tissue—seeds and mature tissue

When growth is not occurring, the plant tissue may be assayed like a test tube or well-stirred beaker. In a stationary (conventional) reference frame the growth velocity terms are zero, so that only the first term on the right hand side of the continuity equation is relevant:

$$ d=\frac{{\partial \rho }}{{\partial t}} $$
(1C)

Equation 1C tells us the deposition rate equals the observed rate of change in concentration. It would be assayed for instance in a spectrophotometric assay by a change in color intensity. In the Bradford assay using Coomassie Brilliant Blue dye, an increase in blue color would indicate protein production; a decrease in blue color would indicate destruction of the protein. Seeds and non-growing tissue producing or degrading protein without changes in volume can be characterized in this way.

One-dimensional, steady cases—growing roots and monocot leaves

To find the spatial pattern of deposition rates within growing tissue we must know not only the time rate of change of the metabolite at locations of interest, but also the tissue expansion rate and the displacement of the tissue relative to the spatial pattern. A one-dimensional example is suitable for many problems involving roots, monocot leaves, or internodes,

$$ d=\frac{{\partial \rho }}{{\partial t}}+\rho \left( {\frac{{\partial {v_z}}}{{\partial z}}} \right)+{v_z}\frac{{\partial \rho }}{{\partial z}} $$
(1D)
$$ \mathrm{Deposition}\;\mathrm{rate}=\mathrm{Local}+\mathrm{growth}\;\mathrm{dilution}+\mathrm{convective}\;\mathrm{rate}\;\mathrm{of}\;\mathrm{change}\;\mathrm{rate}\;\mathrm{of}\;\mathrm{change} $$

This equation is derived from the assumption of conservation of matter (e.g. Boas 2005). Here the deposition rate, perhaps in ng per mm of tissue along a plant axis, is a local rate ascribed to a particular position. The equation is in terms of changes in concentration with respect to time, t, and distance, z. We see that the equation needs data on both concentration and growth. To find the spatial pattern of growth velocity, v z , it is convenient to use the reference frame with origin at a root or shoot apex, or the base of a monocot leaf. That is, z = 0 and v z  = 0 at the origin and v z increases with z to reach a maximum displacement velocity at the base of the growth zone (Erickson and Sax 1956). The velocity gradient is the “relative elemental growth rate” (“REGR”) quantifying the local expansion. In this moving reference frame, in the simplest case, growth patterns may be steady (unchanging in time), while cells are formed and displaced through the growth zone (Erickson and Sax 1956; Boffey et al. 1980). In growing roots and monocot leaves the spatial pattern in mineral content may also be steady (Silk and Erickson 1979). The gradient terms may be evaluated in spreadsheets using Erickson’s midpoint differentiation formulas (Erickson 1976), or cubic splines may be fit to the data and numerically differentiated, for instance using MATLAB (The MathWorks, Inc.).

Figure 1 gives an example showing the data needed to calculate the deposition rate in a root tip growing in steady state. Experiments are needed to determine (left) mineral concentration ,with calculated concentration gradient, and (right) growth velocity, with calculated velocity gradient. Here we find that the local rate of change in mineral concentration is negligible. For instance, at the location three-mm from the root tip we find 112 ng of a mineral element whether we look at a young or older root (Fig. 1). We can thus neglect the first term, the local (site-specific) rate of change in concentration. Although the concentration profile was found not to change in time, the remaining terms of Eq. 1D tell us that mineral deposition is certainly occurring. The second term of the equation, the “growth dilution” term, is the product of the local mineral concentration and the REGR. It is easy to believe that if the tissue is expanding 38 % per h at the location 3 mm from the root tip, then the tissue element must be depositing 38 % of 112 ng, about 42 ng, each hour to maintain the observed local concentration of the mineral element, 112 ng per mm. The last term of the equation, the “convective rate of change,” is harder to grasp. An analogy makes it easier to see as the term needed to maintain an observed gradient in concentration (Fig. 2).

Fig. 1
figure 1

Data to calculate the deposition rate of a mineral element in the elongation zone of a corn root. (Left—content data) One –mm segments are collected by position and assayed for mineral content in young root tips (solid black line) and roots one day older (thinner solid line); the curve of content is differentiated with respect to position (dashed blue line) to obtain the concentration gradient. (Right—growth data) Observation of mark positions at 15 min. intervals permits calculation of growth velocity (solid blue line) and differentiation of the velocity to obtain the velocity gradient (“REGR”) as a function of position. At any given location (the gray vertical line shows the three mm position) the product of values of the components in black are summed with the product of the components in blue to calculate the value of the deposition rate

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Fig. 2
figure 2

Analogy to visualize the concept of the convective rate of change. (Above) Imagine that an artist wants to paint a school of fishes swimming in a stream. The problem is how to maintain the pattern of fishes long enough to paint it. (Below) An answer to this problem is that the artist can hire workers to throw fishes into the stream. The faster the fish are moving, the more rapidly the workers must add new fish. And the greater the difference in fish between locations, the more rapidly the workers must throw in the new fish. This analogy suggests the logic of the convective term, the product of velocity and spatial gradient (change with position) in concentration

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The growth dilution term is always positive in growing tissue. The convective rate of change is negative if the concentration decreases with position (a negative gradient as in Fig. 1), and positive if concentration increases with position so that the tissue element increases in concentration during its displacement (as suggested in Fig. 2). The experimental design used to obtain Fig. 1 gives results per mm of root or leaf. These results can be made more universal if they are divided by the weight or cross-sectional area of the segments used for the chemical assay, to give deposition rates on the basis of tissue weight or volume (Silk et al. 1984).

Sampling and time intervals

The continuity equations show that the spatial patterns of growth velocity and metabolite concentration are the basic data required to find the metabolite deposition rates. Tissue sampling should provide adequate resolution through the growth zone. Indeed linear growth zones show a large range of length. Determining metabolite concentration profiles is easier in long growth zones without needing to pool samples. Recently, Nelissen et al. (2012) used the 65 mm-long growth zone of the maize (Zea mays L.) leaf to facilitate the determination of several hormone concentrations. The ~30 mm-long growth zone of grasses such as tall fescue (Festuca arundinacea Schreb.) leaf offer also a good model (Schnyder and Nelson 1988; Gastal and Nelson 1994). A mm-scale resolution is required for plant roots whose growth zone is around 10 mm such as maize (Sharp et al. 1988) or poplar (Populus deltoïdes × nigra cv. ‘Soligo’) (Merret et al. 2010). The shorter root growth zone of the model plant Arabidopsis thaliana (L.) Heynh. (~2.5 mm, Beemster and Baskin 1998) requires much higher resolution, using microdissection or imaging of quantitative and sufficiently short-lived reporters. At the other end of the size scale, the giant blades of bull kelp (Nereocystis luetkeana (K. Mertens) Postels & Ruprecht) have growth zones almost one meter long and can be sampled on the 5-cm scale (Koehl et al. 2008).

For the growth analysis, the movement of natural or applied marks should be followed on a time scale long enough for mark displacement, but short enough that marks do not move too far from their initial, ascribed positions (Silk 1984). A rule of thumb is that the time interval for observation should be chosen so that marks do not move more than 15 % of the length of the growth zone. To capture the growth near the origin (where growth velocity is small), it may be useful to follow the marks over a longer time scale than that used to analyze the faster moving marks. Ideally, the sampling time scale should be adequately designed to also assess shorter temporal variations-hourly oscillations and circadian rhythms (Merret et al. 2010).

Comparison to radiolabel experiments

The deposition rate computed with data on growth velocity and chemical concentration would be expected to parallel the results of an uptake experiment in which the tissue is incubated with radioactive label. Results are not in general numerically comparable, since the continuity equation returns endogenous deposition rates, while the labeling experiment shows deposition of the exogenous supply. Still, to understand nutrient uptake in roots, a label experiment can give valuable information if an appropriate time scale is chosen for the uptake experiment. Problems in experimental design arise because the radioactive labeling experiment is not instantaneous (Silk et al. 1984). Some time is required for penetration of the label into the tissue, and during this time interval the cells found at one location will be displaced to another. For instance in the corn roots of the study on uronide deposition, cells found initially at 4 mm will be located an hour later at 5.2 mm, where the deposition rate is quite different. As the time interval of the labeling period is increased, the discrepancy increases between the instantaneous deposition rate and the apparent label incorporation rate (Fig. 3). The cells at the sampling locations have arrived from progressively more apical locations, and their label integrates the deposition rates they have experienced in space and time. Numerical experiments show that a 15 minute pulse of radio-label will return a trend similar to a calculated deposition rate (Fig. 3). But a two h pulse will not reveal the peak of the instantaneous pattern, and by five h the incorporation rate pattern appears flat (uniform), in great contrast to the pattern revealed by the continuity equation. The take home message is that the criteria for accurate growth analysis should be applied to a radio-label experiment: the duration of incubation in label should be short relative to the growth displacements of the tissue.

Fig. 3
figure 3

Numerical simulation of radiolabel incorporation experiments. Upper line shows the calculated deposition rate of uronide. Lower lines show profiles of radioactive incorporation that would be apparent after progressively longer periods of incubation in radiolabel. Redrawn from Silk et al. (1984) Uronide Deposition Rates in the Primary Root of Zea-Mays. Plant Physiology 74: 721–726, www.plantphysiol.org, Copyright American Society of Plant Biologists

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Lagrangian (cell-particle-specific) description

Most of the time, concentration and net deposition rate are given as a function of position in the growth zone and describe “what happens” in each particle or cell disposed along the growth zone. It is also interesting to get a temporal and spatial description of these variables in a single particle or cell cohort crossing the growth zone as time goes on (Green 1976; Silk and Erickson 1979; Gandar 1983a, b). This cell-specific description is called a Lagrangian specification as opposed to the spatial (site-specific) or Eulerian specification (Silk and Erickson 1979). If data are given as spatial patterns of a developmental variable, to follow a cell requires determining the Eulerian specifications of growth and of the variable over the time necessary for a particle to cross the growth zone. In non-steady state this requires extensive data on spatial and temporal patterns of the variable and the growth velocity. In steady state (that is, if spatial patterns of both growth velocity and density are not changing with time), the spatial specification can be interpreted as descriptions of “what happens” in a single particle as it is moved through the growth zone. In steady state it is also possible to use a single growth velocity curve to determine the growth trajectory, that is the relationship between particle position and time, from the temporal integration of the velocity field following the particle (Silk and Erickson 1979; Gandar 1983a) or by counting cells outside the meristem (Silk et al. 1989). Then the spatial coordinates can be converted into temporal coordinates to provide the Lagrangian specification.

By displaying the hidden temporal aspects of growth, Lagrangian specifications provide new information. For instance, in maize and pine (Pinus pinaster Aiton) roots, osmotic stress affected relative elemental growth rate pattern (REGR) according to the stress level. It shortened the length of the growth zone (Sharp et al. 1988; Triboulot et al. 1995). Drawing the Lagrangian REGR profiles revealed that duration of longitudinal expansion was similar for all stress levels (Sharp et al. 1988). More recently this approach has been applied to molecular biology. Merret et al. (2010) determined the Lagrangian specification of Actin transcript density (Fig. 4) in roots of poplar. This analysis revealed the long temporal extent of the apical peak and the relatively short duration of the more basal peak. A 3D-plot where both time and space are plotted is another graphical way to show these two points of view (see a rotation 3D-pot at http://www.biomedcentral.com/1741-7007/8/18/additional). In general, determining the Lagrangian specification of transcript densities and gene products is an exciting and unexplored approach for understanding the molecular biology of response to environmental variation.

Fig. 4
figure 4

Actin11 transcript density (a.u. mm−1) as a function (a) of position along the primary root apex of poplar and (b) of time in an element positioned initially at 1 mm and moving across the growth zone (from Merret et al. 2010)

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As applied in Fig. 4, the classical way consists of drawing the growth trajectory of a particle initially positioned near the meristem as it is moved toward the distal positions by the elongation of younger cells. However it is also possible to go back in time and to determine the growth trajectory of a particle that finishes its route at a determined final position. Figure 5 shows the “reverse” growth trajectories of particles spread every 0.25 mm along the growth zone of a poplar root apex (drawn from Merret et al. 2010). The interesting point is that, 20 h before, these particles were all positioned in the segment [1–1.75 mm]. Thus drawing the temporal specification for particles reaching a range of final positions may be also useful for comparing treatments affecting the length of the growth zone.

Fig. 5
figure 5

Reverse trajectories of particles spread every 0.25 mm along the growth zone at t = 0, for the 20 preceding hours (drawn from Merret et al. 2010)

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Biological outcomes: Regulation of metabolite concentration in growth zones

In this section we describe some basic patterns that emerged during early applications of the continuity equation to analyse metabolite production in growing tissue. A later section on biological outcomes then quantifies impacts of environmental variation.

The pattern of metabolite density may be unchanging with time in primary growth zones while the cellular population of the zone is replaced several times

In the 1980s, the Silk laboratory showed that the uronide concentration profile in young maize roots is similar to the concentration profile in roots 15 h older. Figure 6 (redrawn from Silk et al. 1984) shows the uronide concentration profile in the elongation zone above the meristem (1.5–11 mm from the root tip). A separate growth analysis revealed that the cellular population of this zone was replaced twice during the 15 h period. In both young and older roots, the uronide in each tissue element falls from 1.4 μg mm−1 to 0.8 μg mm−1 as the tissue is displaced through the growth zone. In general in roots and monocot leaves the concentration profiles are often quasi-steady, changing only slowly relative to the displacement of the cells through the profile.

Fig. 6
figure 6

Profiles of uronide density in young and older root tips of maize seedlings. Citation as in Fig. 3

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Even when the concentration of a mineral element is decreasing in developing tissue, large deposition of the element is occurring. Growth zones are often the largest sink for metabolites and mineral elements

The Fig. 7a and b show that potassium concentration decreases during root development. This is seen in two different ways. Cells lose potassium concentration as they are displaced through the growth zone—compare [K+] at 2 and 10 mm from the tip. Apparently growth- associated water uptake and dry matter production are greater than potassium deposition, so potassium is diluted as cells become longer. Also, each location has less K+ as the root ages—compare the younger and older roots. And yet the deposition rate profile shows clearly that potassium is deposited rapidly in the growth zone. The rate of 30 μmoles g−1 h−1 in much of the growth zone is 14 times the average rate of K uptake reported for entire young maize roots in the field (Mengel and Barber 1974), and 28 times the average flux into hydroponically grown maize roots (Claassen and Barber 1974). The positive deposition rate indicates the growth zone is a large sink for metabolites. Surprisingly this is true even for potentially toxic heavy metals, copper and zinc (Shi et al. 2009). The large sink activity of growth zones is only apparent when the growth dilution term is taken into account

Fig. 7
figure 7

Profiles of a) potassium density and b) large positive potassium deposition rate and the small component, the local rate of change in potassium. Redrawn from Silk et al. (1986) Spatial Distributions of Potassium, Solutes, and Their Deposition Rates in the Growth Zone of the Primary Corn Root. Plant Physiology 82: 853–858, www.plantphysiol.org, Copyright American Society of Plant Biologists

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Monocot leaves also have developmental gradients and stage-specific patterns of mineral element deposition. Successively initiated maize leaves have somewhat different profiles of mineral elements. Nevertheless it is evident that growing tissue has the most rapid deposition rate and is the strongest sink for mineral elements (Table 1, from data of Meiri et al. 1992). Although local net deposition rates may be slow, continuing addition or removal of minerals may lead to significant local changes over time. The elements Ca and P in particular accumulate in leaf tips of older maize leaves, as indicated in Table 1 (Meiri et al. 1992).

Table 1 Deposition rates of nutrients in developing corn leaves
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Regulation of metabolite concentration depends on net deposition rate, not concentration

In the example above (Fig. 7), an experimenter seeking to understand potassium transport should look for carrier activity or up-regulation of gene expression associated with deposition rate. On the other hand, the physiological importance of potassium (as osmoticum for instance) is related to the concentration itself.

N is deposited rapidly into dividing and elongating leaf cells of fescue; RuBisCO in maturing tissue is formed by cannibalization of previously formed metabolites

Ecologists know that in the biosphere the bulk of fixed nitrogen is found in leaf chloroplasts in the protein of carbon-fixing enzymes—RuBisCO. Molecular biologists are aware of the fundamental importance of nitrogen in nucleic acids—DNA and RNA. Gastal and Nelson (1994) linked these two aspects of the nitrogen cycle when they calculated N deposition rates in growing fescue leaves (Festuca arundinacea Schreb.) (Fig. 8a). Leaf nitrogen concentration was highest in the meristem and elongation zone at the leaf base and fell quickly to a rather constant amount at the end of the growth zone (determined to be about 30 mm long) and in mature tissues. This indicates that nitrogen deposition is rapid in the growth zone and negligible after growth ceases. Nitrate reductase activity (Fig. 8a) and RuBisCO content (Fig. 8b) increased from negligible amounts to a peak beyond the growth zone. Gastal and Nelson thus showed that RuBisCO and other chloroplast proteins in maturing tissue are synthesized from catabolism of compounds formed earlier during cell production and elongation.

Fig. 8
figure 8

a Profiles of leaf nitrogen content in fescue leaves (inset, top right). Deposition rates of total, insoluble, and soluble N and NO3-N and nitrate reductase activity. b Volumetric N deposition rate superimposed on profile of RuBisCO content. (Adapted from Gastal and Nelson 1994, kindly provided by François Gastal)

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Comparison of endogenous deposition rates and exogenous influx reveals contrasting import/export patterns for ammonium and nitrate in growth zones of maize roots

Uptake of nitrogen from soil as either nitrate or ammonium is one of the most important root functions. Bloom and colleagues (Colmer and Bloom 1998; Bloom et al. 2002) measured spatial patterns of uptake of NO3 – and NH4 + from the bathing medium, “exogenous influx,” by observing with ion-specific electrodes the depletion of these ions in the boundary layer next to the root. They found that when nitrogen was supplied as NH4NO3, net influx of NH4 + was rather uniform (~1.4 nmol mm−1 h−1) through the root cap and root growth zone (Fig. 9). When nitrogen was supplied as Ca(NO3)2, the pattern of nitrogen influx was significantly different: net NO3 – influx increased from 1.5 nmol mm−1 h−1 in the root cap to higher values, 5.4 and 7.6 nmol mm−1 h−1 at 3.5 and 11 mm, respectively, from the root apex (Fig. 9). Thus the presence of NH4 + inhibited net NO3 – influx. This inhibition together with the observed pattern of H+ fluxes indicated that the entire maize root apex absorbed more exogenous NH4 + than NO3 − when both forms were present in the medium. This makes sense, as assimilation of NH4 + costs less energy than NO3 −. The root is located far from the photosynthetic organs and is energetically starved relative to the shoot. Assimilation of NO3 − to glutamine expends 12 ATP equivalents versus only 2 ATP equivalents for NH4 + to glutamine.

Fig. 9
figure 9

Figures to demonstrate which forms of N are supplied exogenously from the soil solution and which are transported acropetally from more mature tissues. Exogenous uptake and endogenous deposition rates of nitrate and ammonium into maize roots with (top) calcium nitrate supply or (bottom) ammonium nitrate supply (data from Bloom et al. 2012)

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Bloom et al. (2012) then were able to infer the patterns of import and export of different forms of nitrogen by calculating endogenous nitrogen deposition rates and comparing these to the previously determined exogenous uptake rates (Fig. 9). Where local deposition exceeds local influx, the tissue must be importing endogenous nitrogen. Under Ca(NO3)2 supply, root NO3 – influx was adequate to account for pools found in the growth zone and provided twice as much as was deposited locally throughout the non-growing tissue. Some NH4 + was deposited and must have been imported from the seed or shoot. In contrast, under NH4NO3 supply, NO3 – influx was less than the local deposition of NO3 – in the growth zone, indicating that additional NO3 – was imported or metabolically produced. Yet the profile of NO3 – in the growth zone was similar for the plants receiving Ca(NO3)2 or NH4NO3 (Bloom et al. 2012). For both N supplies, tissue nitrate levels increased from the meristem through the growth zone and all of the apical 60 mm. Xylem sap NH4 + concentrations were indifferent to the presence of NO3 – in the medium. Since the exogenous uptake of NO3 – in the root apex is highly dependent on the presence of NH4 + in the medium, when NH4 + is present, all the NO3 – absorbed near the apex is stored as free NO3 – in the tissues, and additional nitrate is imported or metabolically produced. When NO3 – is the sole N-source, the roots store similar amounts of free NO3 – and they absorb additional NO3 – that is assimilated or translocated to the shoot. These results show that NO3 – deposition, whether exogenously supplied or imported, is important for supporting root elongation in the basal part of the growth zone and maintaining root function in the young mature tissues.

In a related study, Walter and colleagues (2003) determined the endogenous fluxes and deposition rates of metabolites in the seedling roots of maize. They showed that nitrates, but not other nutrients, were exported to a great extent from the root growth zone to the mature tissue. Overall, their results show that a careful analysis of growth kinematics allows quantifying and interpreting a number of important flux parameters in the growing organ.

Biological outcomes: Metabolite deposition rate and growth response to environment

Under water stress, if maize roots do not experience a change in mechanical impedance, osmotic adjustment occurs with decreased rates of solute deposition coupled to larger decreases in radial growth rates

In plants subjected to water deficit turgor may be maintained by true osmotic adjustment, which consists of decreasing cell osmotic potential by increased solute content without change of relative water content. In growing tissue, osmotic adjustment can result from either osmoticum accumulation or reduced tissue volume expansion. In maize seedlings submitted to different levels of water deficit, the osmotic potential along the root apex decreased in proportion to the stress level while relative water content was weakly affected, indicating true osmotic adjustment (Fig. 10) (Sharp et al. 1990). Based on osmotic potential and water content profiles, the net deposition rates of osmoticum were calculated using the continuity equation (Eq. 1C), per unit length and per unit volume. While the net deposition rate relative per mm decreased over most of the root tip in response to water deficit, the net deposition rate relative to volume increased in the apical part of the apex (Fig. 11). This happens because root radius and water content per unit length were strongly decreased by the water deficits all along the growth zone (Sharp et al. 1988, 1990). Thus, the drought-induced decrease of radial root growth allowed substantial osmotic adjustment despite reduced solute deposition and contributed to the maintenance of primary root elongation in the three apical millimetres. This highlights the importance of considering growth in all dimensions (and not only the main strain direction), especially when comparing among treatments. The comparison of the net deposition rate of hexoses, known to be involved in osmotic adjustment, with that of potassium, a major mineral component of cell sap, showed that hexoses contributed to osmotic adjustment while potassium did not: solute net deposition rates were species-dependent and uncoupled from volumetric growth.

Fig. 10
figure 10

Spatial distribution of osmotic potential in the apical 10 mm of roots growing in vermiculite of various water contents. The treatments represent the vermiculite water content as percentages of the water content at high water potential. Osmotic potentials are means ± 1 SD from three to seven experiments. From Sharp et al. (1990) Growth of the maize primary root at low water potentials. II. Role of growth and deposition of hexose and potassium in osmotic adjustment. Plant Physiology 93: 1337–1346, www.plantphysiol.org, Copyright American Society of Plant Biologists

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Fig. 11
figure 11

Spatial distribution of net deposition rates of osmoticum in the apical 10 mm of roots growing at the various vermiculite relative water contents. Citation as for Fig. 10

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Similar conclusions were found in growing leaves of tall fescue after water was withheld (Spollen and Nelson 1994). The increasing levels of water deficit reduced the relative elemental elongation rates and shortened the growing zone in proportion to the duration of water withholding. When the water deficit exceeded a threshold, the dry matter content per unit length was increased by more than 20 % over the whole growing zone while leaf elongation rate was decreased by approximately 50 %, indicating that net dry matter import rate must have been decreased in this growing zone. Further analyses of water soluble carbohydrates content suggested that the hydrolysis of fructan stored in the distal part of the elongation zone could account for an increased content in hexoses in the apical part. It is clear that the accumulation of water soluble carbohydrates leads to osmotic adjustment.

We note that these effects of drought occurred only in the absence of changes in soil hardness. In contrast roots often grow thicker in those soils that become hard to penetrate as they dry. (reviewed in Bengough et al. 2006). Work remains to analyze the growth kinematics and metabolism as the root adapts to increased soil impedance.

In salt-affected sorghum leaves, decreases in potassium deposition rates parallel decreases in growth rates

Bernstein et al. (1995) analyzed growth and mineral deposition rates in leaves of sorghum (Sorghum bicolor [L.] Moench, cv. ‘NK 265’) plants grown in nutrient solution with and without 1,000 mol m−3 NaCl salinity. Salinity affected both growth and mineral content. The relative elemental growth rate and water deposition rates were not much affected in the proximal meristematic regions and were progressively more inhibited toward the base of the growth zone (Fig. 12a). Salinity produced elevated concentrations of Na and decreased concentrations of K and Ca; while Mg deposition rates were synchronized with water deposition rates so that tissue concentrations of Mg were not much affected by salinity. The decrease in Ca deposition rate relative to control was uniform along the growth zone (Fig. 12b). In contrast the K deposition rate was more strongly affected in distal regions (Fig. 12c). The parallel inhibitions of growth and K deposition rate suggest the latter might be particularly important in regulating the growth response to salinity.

Fig. 12
figure 12

Salinity effects on the relative elemental growth rate (top graph), calcium deposition rate (middle graph) and potassium deposition rate (bottom graph) as compared to control. Redrawn from Bernstein et al. (1995) Growth and development of sorghum leaves under conditions of NaCl stress - possible role of some mineral elements in growth-inhibition. Planta 196: 699–705 with kind permission of Springer Science + Business Media

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Aphid infestation of alfalfa transforms apical nitrate sink tissue to a nitrate source

Girousse and colleagues (2005) studied carbon and nitrogen deposition in alfalfa (Medicago sativa L.) infested by a pea aphid. This aphid is a phloem feeder that does not transmit virus or toxins, and assimilate withdrawal is generally considered as the main mechanism responsible for growth reduction. Girousse et al. found that severe short-term aphid infestation (200 young adults over a 24-h period) induced a strong reduction in growth rate, particularly longitudinal expansion (Fig. 13-I). Profiles of carbon and dry matter were unaffected by the infestation, indicating that the growth reductions were synchronized with reductions in deposition rates of water and carbon (Fig. 13-II). However the aphid effects on plant nitrogen were quite different. Reduced nitrogen content and associated negative nitrogen deposition rates (Fig. 13-III) were observed in the apex. This indicated a mobilization of nitrogen from the apical part of the growth zone, converted from a sink tissue into a source tissue by aphids. From the growth anisotropy the authors were also able to conclude that aphid-induced thigmotropism is associated with the long term effects of the infestation.

Fig. 13
figure 13

Effect of aphid infestation on spatial patterns of (I) growth, (II) water, dry matter, carbon, and nitrogen, and (III) deposition rates of water, dry matter, carbon, and nitrogen in alfalfa stems. Adapted from Girousse et al. (2005) Aphid infestation causes different changes in carbon and nitrogen allocation in alfalfa stems as well as different inhibitions of longitudinal and radial expansion. Plant Physiology 137: 1474–1484, www.plantphysiol.org, Copyright American Society of Plant Biologists; kindly provided by Christine Girousse

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Molecular biology: Utility and outcomes

In mature tissues, where cells do not enlarge, the transcriptional regulation of gene expression (e.g., environmental induction) is computed as the temporal change of transcript density in a stationary reference frame. In practice, gene induction or repression is inferred from the ratio of the gene expression in treated plants relative to controls. When focusing on development, gene expression is compared in samples collected in different positions representative of successive time points. For instance, highly accurate maps of gene expression in the root apex have been produced, highlighting coordinated gene expression networks (Birnbaum et al. 2003; Brady et al. 2007). Since longitudinal profiles describe the different developmental stages undergone by a cell during growth, one could be tempted to deduce regulation of gene expression from variations of gene expression along the growing zone. However, this would be misleading since cells expand at varying rates while being displaced along the root. The continuity equation can be used with growth trajectories to get an unbiased view of the spatio-temporal regulation of gene expression.

Actin11 is highly regulated by a balance of transcription and decay during growth of poplar roots

This framework has recently been used to analyze the regulation of gene expression within the growing zone of the root. Merret et al. (2010) determined transcript densities and calculated rates of change in gene expression and molecular developmental information in the poplar root apex. The authors focused on two genes widely used as internal control in real-time qPCR (Brunner et al. 2004), Actin11 and Elongation Factor 1-β (EF1β). Gene expression was quantified at high spatial resolution with real time PCR. The Actin11 transcript density profile displayed two peaks at 1.5 and 3.5 mm from the root tip with similar spans but decreasing intensity, and low values in more distal segments (Fig. 4). The net accumulation rate of transcript reveals the regulation of gene expression resulting from the balance of transcription and decay along the root, corrected for the positional and dilutive effects of growth (Fig. 14a). Positive rates denote up-regulation while negative rates denote down-regulation, it was shown that Actin11 expression profile resulted from highly controlled regulation. The drawing of the Lagrangian specification also highlighted that the high expression level in the segment [1–2 mm] is produced by a quite small but long-lasting induction and that classical spatial specifications hide temporal aspects of regulation (Fig. 14b). This work showed that this gene cannot be used as reference gene for normalization in real time PCR when developing organs are considered.

Fig. 14
figure 14

a Net accumulation rate of Actin11 transcripts as a function of the distance from the root tip (Eulerian specification), expressed in arbitrary unit mm−1 min−1 and calculated using Eq. (1A). b Net accumulation rate of Actin11 transcripts as a function of the time for a particule starting at 1 mm at t = 0 (Lagrangian specification) (redrawn from Merret et al. 2010)

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While gene expression is usually normalized in order to get rid of sampling error and to eliminate differences of biological activity among samples, the continuity equation must be applied to absolute quantities, thus on transcript densities. Since the density of nuclei and presumably the cellular activity are heterogeneous along the root growth zone, net accumulation rates must be compared to the net accumulation rate of a standard entity, either the expression of reference genes or cell density. Merret and co-authors used total RNA content as an estimate of cell density (2010). As expected, the net accumulation rate of total RNA was null, except in the meristem supporting the significance of the conclusions about the regulation of Actin 11 expression (Merret et al. 2010).

A model shows the regulatory importance of hormone dilution by plant growth

Recently, a modelling approach has suggested that growth-induced dilution of gibberellins (GA) can explain the dynamics of cell elongation in the Arabidopsis thaliana root apex (Band et al. 2012). Assuming that all parameters are steady, the model uses cell diameter, cell length profile along the growing zone and root growth rate to describe the volume increase of a growing cell in space and time, that is, in a Lagrangian way. The model also takes into account the biochemical rules determining hormone transfer among four subcellular compartments: the vacuole, nucleus, cytoplasm and cell wall. Based on transcriptomics and reporter analysis, GA biosynthesis and degradation are supposed to be null in the elongation zone. So, the model estimates the GA concentration in the cell during its expansion and predicts a significant growth-induced dilution. In parallel, the concentration of components belonging to the signalling networks, including the gibberellin receptor GID1 and the well-known growth repressing DELLA proteins, are calculated from the corresponding measured mRNA levels and the cell dimensions. Then, the molecular mechanism regulating the amount of DELLA proteins are incorporated in the model (To simplify, each gene family is treated as a single representative species). The multiscale model shows that the declining GA concentration along the elongation zone affects the level of downstream signalling components, in particular increases the concentration of DELLA proteins, which in turn decreases the relative cell elongation, implying a feedback loop. These implications were tested and confirmed through the phenotyping of roots treated with paclobutrazol (inhibiting GA biosynthesis) and GA biosynthetic mutants. This study also reveals that even if DELLA expression level appears constant over the duration of cell elongation, the protein level can vary, controlled by its degradation rate, which depends on the GA concentration. Authors note the importance of growth-induced GA dilution as cells pass through the elongation zone as a key finding in the control of cell elongation.

Deposition rates of GA regulate growth of maize leaves

Deposition rates of gibberellins have recently been shown to have an important role in regulating growth of maize leaves. Nelissen et al. (2012) made direct measurements of endogeneous hormone concentrations in the maize leaf growth zone and showed that the bioactive GA1 peaks in the transition zone at the distal end of the cell division zone. Using the continuity equation they showed that GA1 accumulation rate is positive in the division zone and the proximal part of the transition zone but drops very rapidly to negative values at the entrance in the strictly-elongation zone, highlighting that, in this zone, catabolic activity and/or export take place in addition to dilution. The combined metabolic and transcriptomic profiling revealed that the GA maximum is established by GA biosynthesis in the division zone and active GA catabolism at the onset of the expansion-only zone. Use of mutants deficient in GA and transgenic plants that overproduce GA confirmed that the size of the cell division zone varies with the GA deposition rate.

Fluorescent tagging of cell membranes and nuclei combined with automated particle tracking may greatly facilitate calculation of deposition rates in plant organs

Recently, powerful computational techniques have been disseminated to allow automated particle tracking so that cell cohorts are followed through space and time. An exciting approach, which the authors have called in planta cytometry, allows automated, nondestructive analysis of both growth and spatial patterns of gene expression (Federici et al. 2012). Nuclei were labeled with one fluorescent protein, and membranes were labeled with another; cells imaged with confocal microscopy then had red nuclei and blue edges. Using a particle search algorithm index cells were followed in space and time. Federici and colleagues then wrote an active contour segmentation algorithm that allowed them to quantify cell and shape and hence growth strain rates. These numerical methods are now available for use with the Image J platform (maintained by NIH). Then cell-specific gene expression was quantified by ratiometric measurement of spectrally distinct nuclear fluorescent proteins expressed under the control of differently regulated promoters. This paper implies that automated high-throughput technology is now available to analyze many aspects of growth kinematics in experimental systems that can be genetically transformed. In particular, the coming decade should reveal many cell-specific (Lagrangian) descriptions of deposition rates and inclusion of dilutional and positional effects to identify gene regulation causing organ development.

Conclusions

The continuity equation, used with empirical data on growth velocity and concentration, allows computation of biosynthesis and deposition rates in growing tissue. These deposition rates give important insight into the mechanism of plant response to drought and salinity, and they clarify many aspects of nitrogen metabolism in roots and leaves. This approach is also essential to understand molecular regulation of growth. Models of the molecular biology underlying plant development are still dominated by time-dependent formulations that are inadequate because they neglect cell displacement and expansion. However several pioneering studies have incorporated growth analysis into well-designed experiments to discover the kinematics of gene regulation. And new technologies—in planta cytometry, coupled with powerful new computational routines and real-time confocal microscopy—promise advancement in understanding of gene regulation in growing organs, especially in those systems that can be transformed with fluorescent tags.