The use of linearity and non-linearity in plant growth analysis

Introduction

Plants are known to loose the ability to grow exponentially, see e.g. Paine et al. (2011). Earlier papers showed, see (1) and (2), that plants do not even grow exponentially from day one: There is no linearity between growth rate and plant size. It is one of the reasons for non-linear modelling in plant growth, see again Paine et al. (2011). But linearities have well-known advantages , as

  1. they are more easy to measure.
  2. they are less complex and difficult to understand than non-linearities.

Exponential growth was an easy to understand phenomenon because of the assumed linear relation between growth rate and size of the mass, resulting in the reasoning of first publisher Blackman (1919) analogous to the compound interest theory. Unfortunately for simplicity, it is not happening in a plant, right from day one.

But too soon a focus on non-linearity is not necessary: We found linearities! This paper discusses the experiments and reasoning mentioned above again, but now with focus on the use of linearity and non-linearity.

Linearities

First you have to consider that the concept exponential growth, the linearity between growth rate and mass, is still true for the growth machinery. It implies that for further research you have to split the life of a plant in at least two phases: The growth machinery in the germination phase is the haustorium and in the vegetative phase the foliage. The speed of the mobilization of the seed energy can be different from the speed of  the energy provision by photosynthesis. Measurements of the size of the growth machinery, i.e. total leaf area per plant show that there is still no exponential growth in both phases from the start. The explanation is that the part of the non-growth machinery in the total plant mass increases in both phases. It was possible to quantify this exactly, as both components of the total leaf area, average number of leaves and average area per leaf were linearly related to time, provided you consider both phases (see 3, 4, 5, 6) It implies that the relation between average total leaf area per plant and time, as product of these two linearities, is a polynomial of the second order. (Notice we used advantage 1. of linearity: easy to measure)

Next, a  very simple model based on well-known linear functions is used (see 7) (Notice the use of advantage 2 of linearity: comprehensibility). Two linear assertions were added:

  • the function of the non-growth machinery is always competition (avoidance) ability (understandable from a Darwinistic view that in the struggle for survival nothing can be useless)
  • the difference between the growth of the growth machinery and the growth of the competition (avoidance) ability is constant (understandable if you want the model to match with the observed second order polynomial)

Then you can get:

  • a plant growing according a second order polynomial
  • with a growth machinery doing also, confirming the measurements of total leaf area
  • and the competitive ability machinery also

Notice 1: The derivative of a second order polynomial is a linear function: The growth rate of total plant mass, growth machinery mass and competition (avoidance) ability mass is therefore also a linear function of time, if once again we recognize the different phases. The relationship between them are also linearities.  So, a lot of linearities. It makes the mentioned simple model even simpler and we should look for that to make explanations simpler.

Notice 2: Not all linearities are simple to measure. Plant mass is difficult to determine, because the measurements are destructive. Next, sampling at very short time intervals is required to detect the linearities in both phases. That implies a lot of plants in big experiments, But we should do this in the future to confirm the conclusions. The distinction in growth machinery mass and competition (avoidance) ability is even more difficult to measure because of questions like: Is only the photosynthetic tissue contributing to growth or also the cells in the roots absorbing water and nutrients ? How do you determine their mass?

Non-linearity

The search for linearity was only abandoned when looking for the root cause. Every Darwinist would, considering that the competition (avoidance) ability is making the plant not growing exponential, suggest evolution: The struggle for survival will cause second order polynomial growth. Analysis of this suggestion is, with a lot of factors involved in far from equilibrium processes, typical the area of non-linearity. Simulation is often used to approach the solution in these situations, when the factors ar not known. Preliminary simulation experiments, as simple as possible, with two-dimensional plants showed that second order growth can win from lower and higher order polynomials (see 8, 9, 10).

Notice 3: We did not explain why a plant, growing according a second order does partition its growth machinery in the two measured linear components: number of leaves and average area per leaf (or number of development units and average area per leaf, see seedling stage of soybean). Suggestion: Do an simulation with three-dimensional plants. Maybe evolution makes this necessary, makes it the winning strategy.

Constraints concerning the modelled linearities

The model only works for the acceleration phase(s) of plants. After this phase(s) plants tend to slow down their growth rate for some reasons and show sometimes a period of linear growth before slowing down again to almost growth rate of zero, see e.g. again Paine et al (2011). A sigmoid curve emerges. There are a lot of questions:

  • Linear growth is often, logically, explained for plants in a crop as follows: When the canopy closes, individual plants cannot expand their leaf area and they can only produce at a constant rate, resulting in linear growth. So, are individual free standing plants really showing linear growth? The mentioned experiments did not show it. Notice, the 2-dimensional simulations did not offer the possibility for the plants to grow together in a crop. As soon the plants touched each other, the competition was evaluated and only the winning plant survived. In case of a draw, both died.
  • The stopping of growth is typical for plants who will die after their generative phase, annual and biennial plants. But what do perennial plants? The used data considered mainly annual plants (see again soybean, sun flower, rice, sage 3, 4, 5, 6). The experiment with the only perennial plant, the oil palm, stopped after the (very long!) acceleration phase.
  • Woody perennial plants have a special trick in competition: They make by producing wood a lot of dead tissue, i.e. they don’t spend energy in maintaining it, but at the same time they use it still for competition, i.e. hight, to overshadow other plants. Notice, the model is not providing the possibility to use dead tissue for competition.

Conclusions

Experiments are needed to answer these questions.  But we can conclude: Split plant life in phases, primarly based on the energy provision and distribution, and look in those phases for models, as simple as possible, preferably linear, explaining regulations on an higher level. But non-linearity is certainly needed when we have to explain the emergence of these regulations by evolution.

References

Blackman, V.H. 1919. The compound interest law and plant growth. Annals of Botany, os-33, 353–360.

Paine, C.E.T.; Marthews, T.R.; Vogt, D.R.; Purves, D.; Rees, M.; Hector, A.; Turnbull, L.A., 2011. How to fit nonlinear plant growth models and calculate growth rates: an update for ecologists. Methods in Ecology and Evolution. Volume 3, Issue 2 p. 245-256