Q1 | What method should I use to estimate RGR? |
Q2 | What statistical test should I use to test for differences in RGR? |
Q3 | What is a simple parameter to express how variation in RGR at a growth-analytical level is brought about? |
Q4 | What are reference values for growth parameters? |
Q5 | How can I estimate RGR of Arabidopsis in a high-throughput manner? |
Q6 | Do you have some exercises for RGR calculations? |
All of the following methods suffer to a certain extent from the same problem: not all individuals in a plant population are equal. This implies that there will be variation in weight and leaf area around a theoretical population mean. Your RGR determination will be worst when you harvest only a few plants from a population with a large variability. |
There are two ways out of this problem: 1. Harvest a lot of plants. This is a rather unattractive solution, as harvesting plants is quite boring. |
2. Reduce the variability in your population, if variability in itself is not of interest. Take care that the growth conditions are as similar as possible for all plants; use, if possible, seeds from a small and homogeneous population of plants, and, most important, discard half of the plants grown before the first harvest. Throw away the 25% largest and the 25% smallest plants the day before you want to start the growth analysis, it is worth the effort. Once you have started you cannot do anything, but hoping for the best. See also Poorter (1989) (p239-240). |
Non-destructive growth analysis |
For hydroponically-grown plants, fresh weight of the same individuals can be measured several times during growth. For plants growing with roots in some solid substrate, the increment in leaf area can be followed. This procedure has the advantage that RGR of individuals can be calculated. Disadvantages: 1. Quite often handling of the plants causes growth retardation. Before embarking on a non-destructive growth analysis make sure your species is an exception. 2. LMF, SLA, dry matter percentage and chemical composition of the plants can only be determined for the last harvest. |
Destructive growth analysis |
A. 'Classical approach'. A large number of plants are harvested on only a few occasions. This procedure is especially appropriate if you are interested in an average RGR over one or two intervals of time. But beware! If you wish to determine ULR as well, and plants more than double in weight between the two harvests you may run into problems if the LAR of the first and second harvest are different. Then you have to determine the exact relationship between plant dry weight and leaf area. The only way to get an impression is to harvest a number of plants in between the harvests. Furthermore, you have to realize that in the classical approach, RGR does not fully equal ULR x LAR. |
B. 'Functional Approach'. Only a few plants are harvested each time, but the number of harvests is large. This procedure is especially appropriate if you are interested in the time course of RGR or NAR. A polynomial is fitted through the ln-transformed values of 1) plant weight and 2) leaf area. The first derivative of polynomial 1 gives the time course of RGR. The LAR can be calculated by dividing polynomial 2 by polynomial 1. And ULR is then simply the derivative of polynomial 1, divided by (polynomial 2/polynomial 1). This is a wonderful method, attractive due to its simplicity. It requires only one assumption: that polynomials 1 and 2 adequately fit the data. But that may also bear a nasty problem with it. How sure are you that the polynomials adequately fit the data? A linear equation explains readily 97% of the total variation, but would a quadratic equation not be better? Unfortunately, the two equations may result in quite different time courses of RGR. The value for LAR is even shakier, as you have to deal with the misfit of two equations. And NAR? Definitely, RGR = ULR x LAR with this method, but you can imagine that putting one uncertainty on top of another will not necessarily result in a proper estimate. These problems are worst at the outer sides of the curves, as uncertainty there is higher than in the middle section. A solution may be to fit with a higher-order polynomial, as it fits the data points better. Unfortunately, this does not help you out of problems. RGR and ULR curves, of data fitted with higher order polynomials, frequently tend to make funny up- and downturns at the end of the curve. The best way to make this problem as small as possible is to test statistically whether the data are described significantly better by a 2nd degree polynomial than by a 1st order equation, and whether a 3rd degree polynomial is statistically more adequate than a 2nd order equation. You have to accept that in a number of cases the ln-dry weight data will be fitted by another type of equation than the ln-leaf area data, and the same will apply to different treatments. This is somewhat unsatisfactory, as the small changes in the equation for ln-dry weight or area, may result in large changes in RGR. An alternative is to fit all data with a polynomial of similar order. Apparently, there is no best solution to this problem. |
C. 'A combination of A and B'. The design of the frequent small harvests is used, but RGR and ULR are calculated according to the classical approach. The calculated values of RGR and ULR show large fluctuations, but by fitting a polynomial through these calculated data, short-term fluctuations are avoided, whereas long-term ones are included. This avoids the assumption that the polynomial exactly describes the growth curve of the plant, fundamental to the functional approach. On the other hand, the short-term fluctuations characteristic of the classical approach are not included either. Theoretically, this may be an attractive solution. Smooth but rather flexible curves are obtained, without too strong assumptions, especially as far as NAR is concerned. However, perfect solutions are like specific inhibitors: they do not exist. The fitting procedure is still rather sensitive to variability in the data, especially towards the outer sides of the curve. A trick to minimize fluctuations somewhat is to calculate the classically derived parameters not for adjacent harvests, but over larger intervals of time. In this way, a running average can be calculated, and if one still likes to have a smooth curve, those running averages can be fitted. The larger the amount of harvests skipped, the less fluctuation in RGR will be found. But this applies to the long-term trends as well. In the extreme case, you will end up with an average RGR only, whereas the intention was to find a time trend! The balance between too much smoothing and too much flexibility cannot be determined objectively, so there is scope for some good feeling here. For more info see Poorter (1989). |
D. Richards function A strong advocate for the use of a less flexible, but biologically more relevant function is David Causton from Aberyswyth. He propagates the Richards function, which is a function with 4 parameters, and a plateau at the end. (See, e.g. Causton & Venus 1982, The Biometry of Plant Growth). This function is very appropriate for describing the growth of a particular leaf, as it plateaus quite clearly. But also in simulations of groth experiments with an interest on total plant weight and area, the Richards function performed surprisingly good (Poorter & Garnier 1996). However, it is a complicated function to fit! |
A problem in the evaluation of RGR and NAR derived from a destructive growth analysis is that the plants measured at harvest 1 cannot be measured again at harvest 2. However, apart from an average RGR also some kind of variability around the mean has to be calculated before any test can be carried out. A simple solution is to pair plants of harvest 1 randomly with plants of harvest 2 and calculate a RGR and NAR for each pair. However, there is no justification for this approach. You may have paired the smallest plant of harvest 1 with the largest of harvest 2, thus overestimating variability. A statistically sound method for the classical approach has been described by Causton & Venus (1981), but this works for two harvests and two treatments only. |
In the functional approach, a 95% confidence interval is constructed around the computed curves for RGR, ULR and LAR. Rod Hunt from Sheffield has written a commercially available program for this analysis. If there is no overlap in the confidence intervals for two treatments, it is safe to conclude that they are significantly different. However, as a consequence of the method used, the 95% confidence intervals are small in the middle part of the curves and wide towards the outer sides. Most likely, you will find a significant difference in the middle part of the experiment, but non-significant differences at the beginning and the end. Clearly, such an approach is statistically sound, but does not bear any biological relevance. Moreover, one can only make pairwise contrasts, a statistically dangerous exercise. |
An alternative is to test differences in RGR with an Analysis of Variance, with ln-transformed dry weight as dependent variable, and Time (of harvest) and Treatment (or Species or Both) as independent variables. A difference in RGR will show up as a significant Time x Treatment interaction. Two remarks have to be made. First, an important requirement is that the design of your experiment is completely orthogonal. That is, you have to harvest the same number of plants for all treatments/species combinations throughout the experiment, preferably equally spaced in time. This is not easy, but remember that an orthogonal design is so powerful that it will be worth the effort anyway. Second, in the case of a large number of harvests, the interaction term may turn out to be non-significant, due to the large number of degrees of freedom. That is, the Sum of Squares of the interaction is rather large, but the Mean Square turns out to be small. This is a complication. Another complication is when harvests are not equally spaced in time. Both problems can be solved using orthogonal polynomials. Thus, the interaction SS can be factorized in a linear, a quadratic. etc., orthogonal polynomial, up to as high as the degrees of freedom for the interaction. Now, mostly, the polynomials with an order higher than 2 or 3 are not interesting and can be discarded as irrelevant fluctuations. Quite often it turns out that the linear and/or quadratic one, when considered separately from the rest, is highly significant, whereas the total interaction was not significant. Lies, damned lies and statistics! For more info see Poorter & Lewis (1986). An SPSS programme to break down the Treatment X Harvest interaction with orthogonal polynomials is: |
A simple way to break down the RGR of a plant is by the technique of growth analysis. RGR can then be factorized into a ‘physiological’ (ULR) and a ‘morphological’ (LAR) component. LAR by itself is the product of the leaf area:leaf mass ratio (SLA) and the fraction of biomass allocated to leaves (LMF). It would be useful to have a simple parameter that tells us to what extent the variation in RGR between species (or treatments) is due to variation in any of these underlying parameters. One way would be to compute correlation coefficients between RGR on the one hand, and parameters like ULR and LAR on the other hand. A high correlation coefficient would imply that variation in RGR (in terms of deviation from the overall mean) scales well with variation in, say, deviations from the mean in ULR. |
The use of correlation coefficients has two drawbacks. First, in the case where only 2 species are analysed, the correlation coefficient will always be 1. Second, the correlation coefficient considers relative variation around the mean, but does not take into account the absolute size of the variation. That is, if RGR varies between 100 and 400 mg g-1 day-1 and ULR varies between 10 and 13 g m-2 day-1, correlation between RGR and ULR can be as high as when ULR varied between 10 and 40 g m-2 day-1. However, in the first case NAR would only marginally contribute to variation in RGR, whereas ULR would be the sole component responsible in the second case. To allow for this Poorter & Van der Werf (1998) introduce the ‘Growth Response Coefficient’ (GRC). They define GRC_{X} as the proportional difference in a particular growth parameter X (ULR, LAR, SLA or LMF), scaled to the proportional difference in RGR. In formula: |
GRC will have a value of 1 if the increase in RGR is fully proportional to the change in X. A value of 0 indicates that the ‘change’ in RGR is not accompanied by any ‘change’ in X. Values below 0 and above 1 may occur also, for example if a higher RGR for a given species is accompanied by both a lower NAR and a more than proportionally higher LAR. |
How are GRCs calculated? In the case of two species A and B, each with a given RGR, ULR and LAR, calculation is as follows: |
If the growth analysis has been carried out well, and RGR really is close to the product of ULR and LAR, then the sum of GRC_{ULR} and GRC_{LAR} is 1.0. In the case of a comparison with more than two species, GRCs are calculated as the slope of the regression line fitted through the the ln-transformed ULR or LAR values (dependent variable) and ln-transformed RGR values (indepedent variable). This slope is dimensionless. For more information see Poorter & Van der Werf (1998). |
RGR | 20 - 350 | mg g^{-1} day^{-1} |
ULR | 2 - 25 | g m^{-2} day^{-1} |
LAR | 2 - 65 | m^{2} kg^{-1} |
SLA | 10 - 130 | m^{2} kg^{-1} |
LMF | 0.25 - 0.80 | g g-1 |
A simple way to grow plants is to grow them in pots. However, for RGR measurements, and especially to break down RGR in components, it is imperative to know the biomass of the whole plant. For most plant species grown in pots this is already quite tedious, because you have to retrieve the roots from the soil, and for Arabidopsis, it is virtually ipossible as the roots are so thin. So life is much easier if plants are grown in hydroponics, because you can quickly remove the whole plant. Growth in hydroponics has many advantages, but it also this is not necessarily easy for Arabidopsis... |
So the alternative is to measure leaf area rather than total plant weight. Total area is easily estimated by a digital camera, and a program like ImageJ. When you do the leaf area measurements on different days, you can calculate RGR on a leaf area rather than a total plant weight basis. How different are these two variables? This simply depends on the change in LAR over time: when LAR remains constant than the leaf area:total plant weight ratio remains constant as well, and then RGR on a weight and leaf area basis are exactly the same. However, generally the LAR does not remain the same, but decreases with ontogeny. Fortunately, the change in LAR over time is in most cases not too large, so that the mistake you make is small. The advantage of this approach is that you assess growth non-destructively, so that you can follow growth of an individual over time, which makes the RGR measurement much more precise. The disadvantage is that you do not know anything anymore about the underlying parameters. |
Be aware that you cannot work with too large plants, as the total area of overlapping leaves will be underestimated in pictures taken from above. Moreover, make sure that you measure the plants each day at the same time, as leaf angle will change diurnally, and a higher leaf angle will result in a stronger underestimation of leaf area. |
You can download an Excel file with some exercises related to RGR by clicking HERE. |