On the Structure-Bounded Growth Processes in Plant Populations
Abstract
If growing cells in plants are considered to be composed of increments (ICs) an extended version of the law of mass action can be formulated. It evidences that growth of plants runs optimal if the reaction–entropy term (entropy times the absolute temperature) matches the contact energy of ICs. Since these energies are small, thermal molecular movements facilitate via relaxation the removal of structure disturbances. Stem diameter distributions exhibit extra fluctuations likely to be caused by permanent constraints. Since the signal–response system enables in principle perfect optimization only within finite-sized cell ensembles, plants comprising relatively large cell numbers form a network of size-limited subsystems. The maximal number of these constituents depends both on genetic and environmental factors. Accounting for logistical structure–dynamics interrelations, equations can be formulated to describe the bimodal growth curves of very different plants. The reproduction of the S-bended growth curves verifies that the relaxation modes with a broad structure-controlled distribution freeze successively until finally growth is fully blocked thus bringing about “continuous solidification”.
Keywords
Plants Population Increment model Optimized ensemble structure Growth process Relaxation-frequency dispersion Growth logistics CommunitiesIntroduction
The growth process of plants and plant populations has been studied intensively [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. In any case, stands of plants show sigmoidal growth curves [4]. A key point of a description is to relate this behaviour to the properties of the individuals [13]. Many growth models have been developed [14] some of them typified by the two factors proportional to (1) increasing tree size and (2) the decline of trees resulting from competition, diseases, or other disturbances. Growth-related molecular details have been investigated such as the process of irreversible cell wall extension [15]; however, despite these achievements description of the growth process remains incomplete. The observation that growth curves of plants fit to several general equations [14] is symptomatic of the difficulty to come to an unambiguous characterization [8].
This situation motivated us to apply a previously described increment model [16, 17, 18]. Increments (ICs) are equivalent molecule clusters cells are composed of. A major advantage is that the actual molecular processes of forming an IC need not be discussed since ICs are assumed to be preformed in the environment or in a reservoir. Cell growth via IC absorption can thus be treated like a chemical reaction. Linked in a cell the ICs hold together via iso-energetic contacts. These are so weak that the configuration of the IC piles fluctuates by thermal activation. Growth processes are enabled to adjust optimal conditions. Putting this in concrete terms the fundamental law results that under stationary conditions the reaction–entropy term (entropy times the absolute temperature) of the increments has to match the contact energy. Yet, this balance is realizable only when a cell-size distribution is formed. The configuration of these patterns fluctuates caused by cell growth and division. Cells of different size are thus stirred all the time providing mixing entropy [16, 17, 18]. Hence, the laws of the thermodynamically founded reaction kinetics determine essential properties of the superstructure. Moreover, fitting experimental data with the equations it can be deduced that [16, 17, 18] different cell types exhibit altogether analogous size distributions.
Of course, when an IC is integrated in a cell, the steady-state conditions are disturbed. Optimal growth conditions must be readjusted via broad-band molecular relaxation. Each growing cell within an ensemble emits signals, e.g. by membrane oscillations [16, 19] thus indicating its presence and condition. The superposition of all signals within a population leads to a modulated field typified by a frequency that increases with the cell number. On the other hand, the distribution of relaxation times that determine molecular rearrangements remains constant. When due to rising cell numbers the interval between signals grows shorter than the relaxation time of a given mode, this mode becomes blocked. Starting with the slowest highly cooperative mode, all subsequent growth-relevant rearrangements are frozen. This phenomenon of relaxation-frequency dispersion leads to S-shaped growth curves, irrespective of the underlying genetic background or individual factors. The phenomenon explains that optimal conditions can be achieved only in ensembles below a maximum cell number. This limit depends to a certain extent on the genetically encoded molecular composition and on environmental conditions [16, 17, 18].
The aim of this study is to use the IC model to describe the growth processes and the superstructure development in plants and plant populations. Here, we pay attention to the existence of a network of finite-sized subsystems. The growth of all these subsystems is expected to be controlled by the same logistics because the model should apply to the level of cellular, organismic and population growth. Consistent applications throughout numerous plant species and even in naturally growing multi-species populations underline the usefulness of this characterization.
Environmental or genetic alteration should enforce a readjustment of optimized distributions. Experimental manipulations of these conditions in individuals are predicted to be integrated in the ensemble structure and consequently influence the ensemble’s growth. According to the underlying principles, even the species composition of plant societies for example studied in various tropical rain forests [20] should be optimized patterns. This may help to explain how the superstructure of abundant species is developed.
The Cell-Size Distribution
Depending on intracellular structure fluctuations p seems to adopt values of 0 ≤ p ≤ 3. At p = 3 the intracellular entropy should be maximal. Extremely small values (p ≤ 1) are expected in self-assembled highly anisotropic protein complexes [21].
A Significant Test
Definitions and Relations
The Relaxation Time
The Interrelation Between ω_{c} and t
Symmetries
The Relaxation-Frequency Dispersion
The signal system in a growing plant population embraces numerous contributions [26, 27, 28, 29] including transmembrane receptors, G-protein-coupled signals, communication during cell division, interaction among leaves, pressure variation in liquid cords, stress impulses, interaction of penetrating roots, etc. Until now, descriptions of growth do not directly account for all these phenomena.
The Growth Process
Growth of Individual Plants in a Population
Bimodal Growth
Extensive quantities like the volume or the biomass should increase proportional to the cell number. Hence, defining adequate scaling factors, the growth of biomass or volume of a plant can be described with Eq. 10. The number \( n^{\prime}(t) \) is obtained by plotting \( n^{\prime}_{{}} (\omega_{c} ) \) against t defined in Eq. 4.
Experimental Results and Discussion
The Growth of a Beech Tree
Growth-invariant parameters of the bimodal growth curve of stem volume of a beech tree according to Hozumi [34]
Vegetative | Generative | ||
---|---|---|---|
Doubling time (Eq. 3) | |||
t_{c1} | 10.5 years | t_{c2} | 22 years |
Maximum value of the stem volume (dm^{3}) | |||
Δn′_{c1} | 70 | Δn′_{c2} | 2.000 |
Kinetic factor of the relaxation time (years) | |||
τ_{kin1} | 0.0011 | τ_{kin2} | 0.00055 |
Upper limit of the relaxation-mode spectrum | |||
ln(τ_{0ymax1}) | 4 | ln(τ_{0ymax2)} | 8 |
Vegetative growth (k = 1) shows a small mode-spectrum with an upper limit of \( \ln \left( {\tau_{y0\max 1} } \right) = 4 \) while in the generative regime it is equal to \( \ln \left( {\tau_{y0\max 2} } \right) = 8 \). According to Eq. 4 the ratio of \( {{\ln \left( {\tau_{y0\max 2} } \right)} \mathord{\left/ {\vphantom {{\ln \left( {\tau_{y0\max 2} } \right)} {\ln \left( {\tau_{y0\max 1} } \right)}}} \right. \kern-\nulldelimiterspace} {\ln \left( {\tau_{y0\max 1} } \right)}} = 2 \) means that the width of the cell-size distribution in the generative phase should be twice that of the vegetative regime. This is in line with the finding that the ratio of the doubling times is also equal to t_{c2}/t_{c1} = 2.1 (Table 1).
Let us use the relation \( (\omega_{ck} /\omega_{0} ) = \exp \left\{ {t_{\text{cross}} /t_{ck} } \right\} \) (see Eq. 4) to calculate the growth-invariant signal frequencies at \( t_{\text{cross}} = 65\,{\text{years}} \). In the vegetative regime the value of \( (\omega_{c1} /\omega_{0} )_{{{\text{cross}}/tc1}} = 488 \) again exceeds substantially the value in the generative phase \( (\omega_{c2} /\omega_{0} )_{{{\text{cross}}/tc2}} = 19.5 \). In the generative regime, optimization is thus achieved at relatively low signal frequencies. Extrapolating the calculation with Eq. 10, a beech tree is predicted to be grown-up at 200–250 years which is in good accord with 200–300 years that is found for Fagus sylvatica L. (http://de.wikipedia.org/wiki/Rotbuche).
Hence, structure development and growth dynamics in a growing tree are strictly connected at any time.
Growth of Chenopodium album Plants
In the vegetative growth regime the relaxation-mode spectrum is narrow (\( \ln \left( {\tau_{y0\max 1} } \right) = 4 \)) while in the generative regime we found \( \ln \left( {\tau_{0y\max 2} } \right) = 12 \). This agrees with values as determined in cell cultures \( \left( {\ln \left( {\tau_{y0\max } } \right) = 12.7 :\tau_{y0\max } = 3.3 \times 10^{5} } \right) \) [16, 17, 18].
Growth-invariant parameters of monomodal growth curves in the vegetative regime and bimodal patterns at times beyond the transition around t_{cross} = 55 days found by estimating the biomass of single Chenopodium plants in a population with the density of 400/m^{2} according to Damgaard et al. [7].
Apart from \( \Updelta n^{\prime}_{cc1} \) and \( \Updelta n^{\prime}_{cc2} \) the values of the parameter sets of the individual plants in the vegetative and the generative regime do not change much
No. | Max. biomass | Kinetic relaxation | Doubling time | |||
---|---|---|---|---|---|---|
Δn_{cc1}′ | Δn_{cc2}′ | τ_{kin1} (days) | τ_{kin2} (days) | t_{c1} (days) | t_{c2} (days) | |
9 + 6 | 1,100 | 11,580 | 3.4 × 10^{−5} | 4.5 × 10^{−5} | 6 | 12 |
8 + 6 | 1,100 | 5,877 | 10^{−4} | 4.5 × 10^{−5} | 6 | 12 |
7 + 6 | 1,100 | 2,800 | 2 × 10^{−5} | 6 × 10^{−4} | 6 | 12 |
6 | 1,100 | 2 × 10^{−5} | 6 | |||
5 | 562 | 7 × 10^{−5} | 6 | |||
4 | 228 | 9 × 10^{−5} | 6 | |||
3 | 166 | 9 × 10^{−5} | 6 | |||
2 | 66 | 9 × 10^{−5} | 6 | |||
1 | 41 | 9 × 10^{−5} | 6 |
The growth-invariant parameters (400/m^{2}) of the growth curves (Fig. 5) of Chenopodiumalbum plants
No. | t_{ck} (days) | τ_{kink} (days) | Δn′_{cck} | ln(τ_{0ymaxk}) |
---|---|---|---|---|
T2 | 8.5 | 3.3 × 10^{−3} | 22 | 4 |
T3 | 8.5 | 1.8 × 10^{−3} | 45 | 4 |
T4 | 8.5 | 1.2 × 10^{−3} | 60 | 4 |
T5 | 9.5 | 1.5 × 10^{−3} | 92 | 4 |
T6 (+T5) | 18 | 1.5 × 10^{−3} | 135 | 12.5 |
T9 (+T5) | 18 | 3.3 × 10^{−3} | 200 | 12.5 |
The growth-invariant parameters of the growth curves (Fig. 6) of individuals of Chenopodiumalbum in a population at 3,600 plants/m^{2}
No | Doubling time, t_{c} (days) | Kinetic relaxation factor, τ_{kin} (days) | Max. height, Δn_{cck}′ |
---|---|---|---|
T3 | 8.5 | 6 × 10^{−3} | 47 |
T5 | 10.5 | 5 × 10^{−3} | 100 |
T9 + T5 | 19 | 5 × 10^{−3} | 205 |
Finding the same growth logistics in the vegetative and in the generative regime does not explain the transition. Well-directed genetic factors seem to be necessary. Not before the vegetative growth is totally blocked [\( \left( {\Updelta n^{\prime}_{cc1} } \right)_{\max } = {\text{const}} \)] does the flowering-period genes [27] succeed in activating this transition. The intersecting lines in Figs. 3, 4, 5, 6 are inserted to highlight the similarity of the transition in vastly different plants like trees and herbs.
Symmetries
Figure 4b (Table 2) shows that in the vegetative regime all data fall in one master curve. Together with the pattern derived in Fig. 4c striking similarities are evident: since the doubling times are the same, the signal frequencies ω_{c1} (Eq. 4) are also identical. Despite different numbers of subsystems [13] all these plants accomplish vegetative growth at \( t_{\text{cross}} \cong 55\;{\text{days}} \). The relatively sharp bending at the end results from the narrow relaxation-mode distribution \( (\ln (\tau_{0y\max 1} = 4) \). Hence, all growth curves can be fitted de facto by exclusively adjusting \( \Updelta n^{\prime}_{cc1} \) (Table 2).
An Important Example
Figure 8b illustrates that the width of the normalized distributions \( n(\eta ,p)/n_{\max } \) increases while p grows, indicating that the mixing entropy within the population increases.
Chenopodium album
Since an optimized superstructure is found in C. album diameter distributions, fluctuations and communication must be present in the population. Of course, aberrant data points come about when plants are clamped in unfavourable configurations. The linear deviations from the ideal state of reference are typified by <γ> = 0.2–0.4.
dbh Distributions
During secondary growth, woody stems show cambium-mediated radial expansion. Recently, the importance of transcriptional regulators, phytohormones and cell wall synthesis in secondary growth was demonstrated [37]. The formation of increments includes these processes.
dbhs are established growth parameters for trees: within a stem, the cylindrical cambium layer induces lateral broadening via cell multiplication accompanied by irreversible cell elongation into the longitudinal stem axis. A fraction of differentiated wooden cell layers develops while the bark is continuously reshuffled. The organisation within a stem runs fast compared to processes at the population level. Lateral growth can thus be described in terms of IC model. The dbh is consequently defined by \( dbh \cong y \) whereby y is the number of ICs. Individual growing trees constitute a forest with an optimized superstructure.
dbh-Frequency Distributions in Beech, Spruce and Pine Forests
Frequency Distributions of 2-Year dbh-Rates in the Old Natural Tropical Foothill Rain Forests, Gajbuih and Pinang Pinang
The fit curves calculated with Eq. 1 and p values around two reproduce the observed distributions fairly well. The dbh growth rate and the absolute dbh classes are linearly correlated: during the relatively short span of growth, the initially optimized superstructure undergoes an affine transformation. The mean deviations induced by aberrant data points are <γ> = 0.20.
The Uniformity of the 2-Year dbh Growth Rates of 12 Abundant Species in a Tropical Rain Forest
Growth-invariant parameters of the dbh increments as depicted in Fig. 13a
System | Δξ (2 years) |
---|---|
Myrsine seguinii | 90 |
Litsea aucuminata | 76.5 |
Illicium anisatum | 60 |
Podocarpus nagi | 54 |
Eurya japonica | 90 (γ = 0.1) |
Distylium racemosum | 48 |
Growth-invariant parameters of the dbh increments as depicted in Fig. 13b
System | Δξ (2 years) |
---|---|
Neolitsea aciculata | 45 |
Symplocos glauca | 31.5 |
Symplocos tanakae | 31.5 |
Cleyera japonica | 18 |
Rhododendron tashiroi | 18 |
Camellia japonica | 13.5 |
Apparently, in warm-temperature rain forests [39] complex modes of interaction organise the 12 species into a superstructure whose members all show conformal distributions.
Final Comments
\( < p > ,\, < \beta \Updelta u_{0} > \,{\text{and}}\,\Updelta n^{\prime}_{cc1} \) of the systems studied here
System | <p> | <βΔu_{0}> | <γ> |
---|---|---|---|
Diameter | |||
Coleus, cortex | 1.88 | 0.10 | 0.21 |
Coleus, outskirt | 1.85 | 0.042 | 0.25 |
Chenopodium | 2.02 | 0.024 | 0.33 |
Spruce | 2.06 | 0.17 | 0.01 |
Pine | 2.27 | 0.22 | 0.29 |
Beech | 1.7 | 0.12 | 0.43 |
Gojabuih | 2.06 | 0.29 | 0.2 |
Pinang Pinang | 1.8 | 0.28 | 0.37 |
Eurya japonica | 2.9 | 1.61 | 0.11 |
Average | <2.06> | <0.32> | <0.26> |
Height | |||
Chenopodium | 0.87 | 0.065 | 0.38 |
During growth, the dynamics freezes, causing solidification as a consequence of relaxation-frequency dispersion. Maintaining stationary growth conditions requires perfect communication among the constituents at all levels, molecules, cells, tissue and organisms. Essentially the strikingly uniform growth-rate distributions of 12 abundant species in an undisturbed natural biotope point to highly cooperative modes of interaction that are not yet understood. Of course, deviations from the ideal line of growth are inevitable. Broad-band relaxation processes compensate for these “defects”.
Objective evidence of a decisive influence of genetic factors is obtained from the description of bimodal growth curves: when vegetative growth is blocked, the coordinated activation of flowering-identity genes [27] is necessary to initiate the transition into the generative regime.
All in all, interpreting growth as an incremental process ruled by thermodynamics in both individuals and entire populations shows plant societies sharing their environment coordinately. In presence of unscheduled but perpetual constraints, resulting from genetic or exogenous causes, optimization at a modified state of reference allows even damaged plants to integrate into the whole ensemble. The community, in return, can readjust an ideal global configuration perhaps by exploiting unused resources. The good correspondence of observed patterns with the calculations indicates ecological integrity.
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