Logistic Models for Simulating the Growth of Plants by Defining the Maximum Plant Size as the Limit of Information Flow

ABSTRACT

Today, the Logistic equations are widely applied to simulate the population growth across a range of fields, chiefly, demography and ecology. Based on an assumption that growth-regulating factors within the Logistic model, namely, the rate of increase (r) and carrying capacity (K), can be considered as the functions reflecting the combination of the organism- and environment-specific parameters, here, we discussed the possible application of modified synthetic Logistic equations to the simulation of the changes in (1) population (density per volume) of photosynthetically growing free-living algae and (2) size (mass per individual) of higher plants, by newly composing r value as a function reflecting the photosynthetic activities. Since higher plants are multi-cellular organisms, a novel concept for the carrying capacity K must also be introduced. We brought the a priori assumption that information sharing amongst cells strongly influences the physiology of multi-cellular structures eventually defining the maximum size of plants, into view. A simplest form of ‘synthetic organism’ conformed to test this assumption is a linear chain of cells, and the first physiological phenomenon, modeled in this way, is growth. This combination of information flow along a chain, with exponential growth, produces a simple allotropic relationship. This relationship is compared with results for plants and is found to have excellent predictive power. This theory shows that fast-growing organisms, or multicellular structures, remain small, because of their inability to share information sufficiently quickly and, also, predicts determinate growth. The success of this simple model suggests, firstly, that the inclusion of information flows in theoretical physiology models, which have been, to date, dominated by energetic or metabolic assumptions, will be improved by incorporating information flows. Secondly, the application of more complex information theories, such as those of Shannon, to biological systems will offer deep insights into the mechanisms and control of intercellular communication.

Introduction

To date, the Logistic function or Logistic curve analyzes are widely applied to the cases in interdisciplinary fields, such as statistics, demography, ecology, geoscience, bio-mathematics, chemistry, economics, sociology, political science, mathematical psychology, probability models, computing, and artificial neural networks.¹ Historically, Logistic models date back to the work by Pierre François Verhulst,²who proposed a self-limiting population growth model with a function or curve expressing a sigmoidal “S-shaped” curve, with the standard equation,

$$f\left(x\right)=\frac{m}{1+e^{-\alpha \left(x-x_0\right)}}=\frac{m}{1+\frac{1}{e^{\alpha \left(x-x_0\right)}}}$$ 1

where values of x maximally range between −∞ and +∞ of real numbers, the midpoint in sigmoid curve is defined by x₀, m is the maximum value, and α defines the curve’s steepness.² The equation proposed by Verhulst is a specific case derived from Bernoulli’s differential equation. The name “Logistic function” which has nothing to do with military or industrial Logistics, was given by Verhulst,³during a study on population growth. By definition, a Logistic curve tends to approach maximal value (m) as the given x approaches +∞ and it approaches 0 as x approaches −∞. Therefore, the initial mode of increase is likely exponential and saturation might be gradually attained in the later phases.

In 1912, microbiologists found that the Logistic equation could be applied to the analysis of bacterial growth patterns under controlled culture conditions.⁴ After its application to human demography in the United States,⁵ Lotka⁶ and Voltera⁷ refined the Logistic model and formulated the following differential equation:

$$\frac{dN}{dt}=r\left(\frac{K-N}{K}\right)N$$ 2

where N is the size of the population, t is time, r is a constant defined as the rate of growth, and K is the carrying capacity reflecting the number of people, other living organisms, or crops that a region can support without environmental degradation. Using several modified Lotka-type equations, Gause,⁸ documented the competition for survival among several Paramecium species.

Note that most living organisms showing a Malthusian-type burst of population growth reflected by the dynamic increase in the number of individuals (N) with the intrinsic rate of natural increase (r) over time (t), soon followed by saturation defined by the environmental capacity known as carrying capacity (K). However, environmental conditions may attenuate the carrying capacity K, thus, it should sometimes be considered as a time-varying factor.⁹

Based on the assumption that factors within the Logistic models are functions reflecting the combination of the species-specific growth characteristics and environmental parameters, the aim of the present perspective paper is to bring the discussion on the possible applications of modified Logistic equations to the simulation of two distinct groups of photosynthetic organisms with distinct dimensions, by focusing on the population (density per volume) of the free-living algae and the size (mass per individual) in higher plants, through newly defining the photosynthetically driven r. In addition, a novel concept for the carrying capacity K must be introduced since the higher plant species form a typical group of multi-cellular organisms. The first half of the present article is dedicated to the description of a photosynthetic formula to be integrated into synthetic Logistic models. A realistic photosynthetic formula must be a function of lighting quality, intensity, and duration, and/or other environmental factors, chiefly, temperature and CO₂ supplies.

In the second half of the present article, we propose the idea that the size limits (K for individual size) for multi-cellular plant individuals under ideal growth conditions could be defined as a series of functions developed with the aid of information theory. By introducing the concept of information to the regulation of growth within multi-cellular organisms, the newly proposed Logistic model for simulating the size of living plants could be a “truly” logistic.

Carrying capacity (K) and growth rate (r) are altered by environmental parameters

Takaichi and Kawano,¹ have previously discussed that habitableness of a given environment harboring living organisms of interest under complex conditions can be evaluated with a modified Logistic model using some model organisms.¹ In general, the Logistic model could be applied to perform simulation of the temporal changes in the population of the organisms of interest (fixed target) under static conditions where environmental parameters are also fixed. Therefore, K and r used in Lotka’s equation are likely empirically fixed.

We have shown that drastic changes in environmental parameters may alter either or both of K and/or r values in Lotka’s Logistic equation.¹ In the case of chemical pollution of environments elevating eco-toxicity, acute toxicity inevitably accompanies the immediate shrinkage of K value. If the altered K is smaller than the size of original population, enforcement of the down-sizing of the population such as induced death or outwards-migration must be observed. Then, K value may remain lowered unless the polluting chemicals are removed or tolerant strains replace the dominant position. Under such conditions, the apparent r value is also minimized. If the toxicity is of chronic mode and the change in K is negligible, there would be certain changes in the r value as demonstrated in aquatic microbes such as green paramecia^10-13 and other Paramecium species.¹⁴

Apart from the fluctuation of the externally determined environmental factors, Yukalov et al.,⁹ have shown that the carrying capacity at time t (K_t) can be expressed as a function of the initial or earlier population, since the population itself size-dependently modifies the surrounding environments. Therefore, carrying capacity or limit of the population size could be considered as a time-varying factor as the following modified Lotka’s equation shows (where K_t > 0):

$$\frac{dN}{dt}=r\left(1-\frac{N}{K_t}\right)N$$ 3

The time-varying nature of carrying capacity was also found in our experimental model using aquatic algal species (data not shown). We are currently demonstrating that the factors determining the growth of algal and paramecium populations, such as r value and K value, can be drastically altered in the model biotopes mimicking the natural environment, depending on the seasonal changes in temperature, shading ratio, mineral composition and outbreak of other interacting microorganisms. Thus, there should be annually or seasonally patterned periodic changes in carrying capacity with period T:

$$K_{t+T}=K_t$$ 4

in which population at time t (hereafter referred to as N_t), is likely determined as a periodic function of period T.

Minimal experiments and modified logistic models for allowing the simulation of the growth of aquatic microbes in the open ecosystems

Since microbial growth in the open eco-system could be simulated with Logistic equation, we have developed the experimental procedures for helping field-based researchers determining the environment-sensitive K and r values which are critical for simulating the growth under a given aquatic condition reflecting the geological, limnological, seasonal, and/or climate variations.

By packing and maintaining the microbes of interest (in our case, two paramecium species) at relatively high density in a capsule allowing the exchange of dissolved gas, inorganic nutrients, and food bacteria but not allowing the entry of wild eukaryotic microbes, a rapid decrease in the paramecium population followed by convergently attaining the sustainable density (N_sus) within a short period (in a week in case of green paramecium), which could be the maximum population possibly maintained in the given environment.

Assuming that population at time t (hereafter referred to as N_t) following the inoculation with higher or lower density of cells (N₀) attains the equilibrium population nevertheless the size of r being whether positive (increasing) or negative (decreasing), and therefore, N_sus could be equivalent with the carrying capacity K, thus,

$$\underset{t\to \mathrm{\infty }}{lim}{N}_t=N_{sus}=K$$ 5

Similarly, the intrinsic rate of increase (r) allowed in a given environment can be determined with model experiments onsite using the above-mentioned capsules immersed in the aquatic environment. By starting the culture in the capsules with N₀ at a low initial density of microbial cells, on the j^th day of culture, we will observe the increase in population (N_j); thus, the rate of increase r can be expressed as follows:

$$r=\sqrt[j]{\frac{N_j}{N_0}}-1$$ 6

These simplified procedures are highly useful for manual determination of K and r values required for simulation of microbial growth, simply by counting the initial and eventual densities of microbes in two distinct model assays with high and low initial microbial densities.

For practical purpose to simulate the population of the microbe of interest at time t (N_t), the Logistic equation was modified as follows, by replacing K and r with experimentally determined values:

$$N_t=N_0+\sum _{t=1}^n\left(\sqrt[j]{\frac{N_j}{N_0}}-1\right)\left(\frac{N_{sus}-N_{t-1}}{N_{sus}}\right)N_{t-1}$$ 7

This formula is routinely applied in our group for daily field surveys for the elucidation of seasonal changes in environmental capacities for microbial growth onsite. When required, following equivalent form of equation which is derived as the solution to the common Logistic equation, is used.

$$N_t=\frac{N_{sus}}{1+\left(\frac{N_{sus}}{N_0}-1\right)e^{-\left(\sqrt[j]{\frac{N_j}{N_0}}-1\right)t}}$$ 8

In addition, by elucidating the doubling time (t_d) and the time constant (C_t) required for attaining the 1/2 N_sus, from the preliminary determined N₀, N_sus and r, the growth curve could be reproduced by the Hill-type (Michaelis-Menten-like) kinetic equation as follows:

$$N_t=N_0+\frac{\left(N_{sus}-N_0\right){\left(t/t_{\mathrm{d}}\right)}^2}{C_t^2+{\left(t/t_{\mathrm{d}}\right)}^2}$$ 9

Note that the maximal value defined in the Hill-type equation (N_sus) and K in the Logistic models are equivalently reflecting the equilibrium population of the organisms of interest.

Conversion of carrying capacity into the maximum size of tissues

In most demographical, ecological and microbial cases, Logistic model handles the density of individuals, colonies, or free-living cells of organisms per area or per volume of a habitable environment, therefore carrying capacity K is defined as the density with square or cubic dimension.

Apart from the concept of density, occasionally, the carrying capacity K in the oncological Logistic model can be defined as the mass of tissue thus predicting the tumor growth and the impact of its chemo-therapeutic elimination.¹⁵ By this way, therapeutic assessment of the tumor size under chemical or physical treatments can be achieved by simply replacing N with X(t) which is the size of the tumor at time t, thus, its dynamics can be expressed as follows:

$$X^{\prime }=r\left(\frac{K-X}{K}\right)X$$ 10

If therapeutic treatments (mostly chemical) are applied, the equation can be modified as below:

$$X^{\prime }=r\left(\frac{K-X}{K}\right)X-c\left(t\right)X$$ 11

where c(t) is the therapeutically induced cell death rate. In this case, the carrying capacity K which is the maximum size of the tumor must be empirically determined.

Kinetic model for simulating the plant size as a function of photosynthesis

For assessment of photosynthetic capacities in a wide range of photosynthetic organisms from algae to vascular plants, photosynthesis-irradiance (PI) curves or light response curves, which are empirical representations of photosynthetic velocity (P) in relation to the intensity of solar irradiance (J) plotted on a graph, are frequently made. On a PI curve, two distinct points of J, namely, the compensation point and the saturation point can be graphically elucidated.

In most plants, there is a roughly linear relationship between P and a given J under relatively low range of J. However, at higher J, most plants hardly sustain the increase in P proportional to the increase in J. Therefore, the rate of increase in P declines as J approaches the maximal level. Historically, there have been several independent approaches for the mathematical handling of PI curves by focusing on the hyperbolic nature of the PI responses in most plants.

Tamiya’s equation (shown below) may be one of the earliest models originally proposed for quantification of photosynthetic assimilation (A) as a function of irradiation (J), while considering respiratory loss (R) and two constants (a and b) to be empirically determined.¹⁶ Tamiya’s constant b represents the slope of P along with a relatively low range of J and the ratio of two constants, b/a, corresponds to the saturated level of P (P_max).

$$A=\frac{bJ}{1+aJ}-R$$ 12

Michaelis-Menten equation (MME) was originally designed for simulating the velocity of enzyme reaction (V) as a function of substrate concentration ([S]), involving two additional factors, namely, V_max and K_m representing the maximum V and Michaelis constant¹⁷ as follows:

$$V=\frac{V_{max}\left[S\right]}{K_m+\left[S\right]}$$ 13

Two British marine ecologists^18,19 have documented their evaluations of several candidate photosynthetic equations and selected one adapting MME as follows:

$$P=\frac{P_{max}J}{K_j+J}$$ 14

In the above Platt-Jasby equation (PJE), P is gross photosynthetic velocity, P_max is the maximum P, K_j is a constant which is the light intensity at which P stays at 1/2 P_max. Net photosynthesis (P_n) can be worked out by subtracting respiration (R) from P (measured as either O₂ evolution or CO₂ uptake by plants or algae). Thus, P_n should be expressed as follows:

$$P_n=P-R$$ 15

Therefore, PJE can be modified as below.

$$P_n=\frac{P_{max}J}{K_j+J}-R$$ 16

Today, MME-derived PJE remains the standard for generation of PI-curves.^20–22

Science history tells us that 3 years prior to proposal of MME,¹⁷ similar but much more powerful equation known as Hill’s equation (HE) has been proposed,²³ to describe the equilibrium relationship between O₂ tension and the saturation of hemoglobin and the concept can be generalized as follows:

$$y=\frac{y_{max}{x}^{\alpha }}{c^{\alpha }+x^{\alpha }}$$ 17

In fact, derivatives of HE are suitable for fitting the sigmoidal curves and they are amazingly applicable to a vast range of kinetics.^1,24 For an instance, MME is a specific form of HE in which Hill’s coefficient $\alpha$ is set as 1.0. It could be also considered that Langmuir adsorption isotherm²⁵ as a modified case of HE.

In addition, kinetic evaluation of drug actions against muscle contraction,²⁶ receptor occupancy theory,²⁷ control of ion channels by ligands,^28,29 and drug regimen control³⁰ have been benefited by modified HE models.

Moreover, informatics can be the target of HE applications as HE has been applied to probabilistic studies with bivariate and multi-variate modifications assessing the risk with multiple factors,³¹ and Shannon entropy and Fisher information matrix in theoretical informatics could be calculated using a HE-derived model.³²

By definition, PJE is also a specific case of HE modifications introducing Hill’s coefficient α=1. By designing the steepness of the curves with altered α, PJE can be rewritten as follows:²²

$$P_n=\frac{P_{max}{J}^{\alpha }}{K_j^{\alpha }+J^{\alpha }}-R$$ 18

This powerful photosynthetic equation should be a composite of photosynthetically driven r in the newly composed Logistic equations for algal and plant growth simulation as discussed below.

Simulating the growth of plant mass based on photosynthetic parameters

Here, we propose the combined use of photosynthetic equations and ecological population models for clarifying the photosynthesis-dependent changes in plant size and population size. According to the discussion by Turchin,³³ ecologists have been collectively using the general law-like principles to guide the development of population models and experiments since the days of Lotka, Volterra, and Gause. Today, the Logistic function or Logistic curve analysis pioneered by these early researchers are widely used in a variety of interdisciplinary fields.

For plant growth simulation, the Lotka-type differential equation can be rewritten as follows, by substituting N with the plant mass (M) and K with the maximum plant size (M_max) which is allowed to attain under a defined condition.

$$\frac{dM}{dt}=r\left(\frac{M_{max}-M}{M_{max}}\right)M$$ 19

Since the rate of plant mass growth (changes in both the fresh and dry weights) must be basically proportional to the net photosynthesis under a changing temperature, water status, nutritional supply, and atmospheric gas composition, especially in the vegetative phase of growth, the photosynthetic status can be a key factor determining the growth rate r. Furthermore, the assimilation efficiency, which is the efficiency with which the photosynthesis leads to cellular growth, may differ from species to species. Therefore, r can be replaced with a photosynthetic parameter (ϕ) and assimilation efficiency factor (ϵ); thus,

$$\frac{dM}{dt}=\varphi ϵ\left(\frac{M_{max}-M}{M_{max}}\right)M$$ 20

In fact, our ongoing research projects handling microalgae and leafy vegetables have revealed that both ϕ and ϵ are sensitive to a temperature within the range of temperatures in a temperate climate. In general, ϕ tends to be greater at higher temperatures and ϵ tends to be maintained at a high rate under low temperature (>0°C). By this way, the kinetics of photosynthesis enables the simulation of the increase in plant mass.

By defining M₀ as the initial size of seedlings planted in a field, the above-mentioned differential equation can be solved as follows:

$$M=\frac{M_{max}}{1+\left(\frac{M_{max}}{M_0}-1\right)e^{-\varphi ϵt}}$$ 21

This equation allows the simulation or prediction of the size of plants at a given t along with a specific environmental history. For practical applications, ϕ can be defined as

$$\varphi =\frac{{\gamma }_{\left(J,T\right)}-{\rho }_{\left(J,T\right)}}{P_{max}^{\ast }}$$ 22

where P*_max is the recorded P_max at a given temperature at which plant or algal species of interest optimally or naturally grows, γ is the gross rate of photosynthesis, which is a function of light intensity (J) and temperature (T), and ρ is the sum of background and light-induced respiration, which is thus also a function of J and T.

The above growth-simulating equation and its derivatives are currently being applied in plant and algal production industries, enabling the precise estimation of the biomass yield under seasonally, daily, and hourly changing climate/meteorological factors.

Theoretical significance of the carrying capacity defining the maximal plant size

Plant biologists nowadays explain the biological phenomena based on the manners in chemical biology and biophysics since the biological phenomena obey the laws of physics and chemistry, thus emphases are on understanding of the nature of “materials” consisting the cells within plants and the status and flow of “energy” captured and utilized by living plants.

Claude Shannon, a father of digital science, has predicted and emphasized that the basis of the universe may not be limited to the energy and the matter, but the information should be considered as the third elemental basis.³⁴ By analogy, there would be an emerging novel concept supplementing the physical and chemical basis for biological phenomena.

As discussed in the cases of paramecium growth, the carrying capacity K (or some equivalent measure such as N_sus) is likely altered depending on the combination of environmental parameters and species. By setting under the optimal environmental parameters, organism might attain the genetically defined maximal size of the population. In the following section, we view that the factor analogous to carrying capacity limiting the plant size can be determined through the novel concept of information flow.

Simulating the growth of plant mass based on information dissemination

Wallbridge and Plummer (unpublished results) are proposing the a priori assumption that flow or exchange of information consisted of chemical and/or electric signals shared amongst cells strongly influences the physiological status of growth and size within the multi-cellular structures. The simplest form of stable multi-cellular structure is the filament or chain of cells.³⁵ We have modeled a simple chain of cells as information units which pass a message sequentially at a constant dissemination speed k (cells per second). If we assume that the chain consisted of cells (sized N) is growing, by cell division, with a doubling time (t_d), then the mean time between cellular growth events is t_d/N, which gets shorter as the chain extends. If the chain continues to grow, it will eventually reach a length (sized N_max) where the frequency of growth events exceeds the ability of the chain to communicate. Since uncontrolled growth often endangers the organisms, we can equate these relationships, at equilibrium chain length, we get a simple result:

$$\begin{array}{c}{N}_{max}={\left(k{t}_d\right)}^{0.5}\mathrm{\#}\end{array}$$ 23

The doubling rate can be replaced with a growth rate μ= ln(2)/t_d to give:

$$N_{max}={\left(\frac{ln\left(2\right)k}{\mu }\right)}^{0.5}=0.83k^{0.5}\mu {}^{-0.5}$$ 24

Note that this result is an entirely theoretical model of information agents, arranged in a chain. There are no assumptions about the mechanism of communication, except that it has a constant rate of propagation along the chain. Nor are there any assumptions regarding any reason for communication. Here, Equations 23 and 24 make clear that as a chain grows, it can reach a point where the communication rate will fall behind the growth rate: thus, the message will never catch up with the growth. At this point, communication along the entire chain becomes impossible, although subsets of the chain may still work for local purposes. Not only does the chain have a maximum useful length but there is also a trade-off between the rate of growth and the maximum length. A chain that grows more slowly, can grow longer, even if the rate of signal propagation is the same as in a faster-growing chain.

Plants certainly have chains of cells which communicate information.³⁶ The vascular cells of the phloem and xylem are adapted to provide fast communication throughout a plant, partly, by being far longer than other plant cells. But the lengthening of these cells, and therefore the variation of cells along the length of the chain, adds at least one new variable. Calculation of the number of cells in the longest chain between the leaf tip and the root tip requires, at least an estimate of, the average cell length (see Table 1 summarized from various sources).

Table 1.

Example of cell size parameter variations in higher plants.

Plant species	Observed maximum growth rate μ (sec−1)	Observed maximum chain length Nmax (cells)	Implied propagation speed k (cells/sec)
Duckweed	2.67E-06	2200	18.67
Arabidopsis	2.67E-06	2267	19.82
Wheat	2.01E-06	2000	11.57
Sunflower	6.69E-07	5417	28.30
Eucalyptus	7.33E-09	65076	44.76
Pine Tree	1.10E-8	16000	4.06

Figure 1 includes results for six plants of widely varying size. The best fit line is N_max = 4.4417 μ^−0.48 (R² = 0.9139). The average message propagation speed (Table 1) is k= 16 cells per second, and the exponent is 0.48, close to the predicted 0.5. The experimentally measured speed of propagation of an action potential in Arabidopsis is between 16 and 26 cells per second,³⁷ which is in good agreement with the 19.82 cells per second predicted by theory.

Figure 1.

Maximum chain length in a number of cells from root tip to leaf tip against growth rate.

These results confirm the suggestion of theory that slow-growing plants can grow bigger than fast-growing ones. It is commonly observed that individuals in many species stop growing, known as determinate growth, when they reach a certain size. This is not for the want of additional energy or nutrition. No matter how rich the soil, benign the environment and plentiful the fertilizer, the plant will stop growing. The suggestion from this research, is that the plant needs to share information around all its cells and, when it reaches a certain size, it can no longer do so.

Conclusions

Growth of plants can be simulated using a newly proposed Logistic model integrating photosynthetic kinetics in the growth rate r. In this novel Logistic model for plant mass growth simulation, the carrying capacity K was rephrased as M_max which is defined as the size limit which is supposed to be determined empirically.

Notably, there would be a hope for theoretically explaining the species-specific plant size limit to be integrated in the Logistic plant growth simulation. A basic model of information flow in a chain of cells has been used to produce theoretical insights into biological behavior. Very simple assumptions of architecture were combined with growth rates to produce a theory of biological growth which has excellent predictive power of the maximum scale of simple biological structures such as filamentous bacteria and more complex multi-cellular organisms such as plants. It is shown that large multi-cellular structures can only grow to a large scale when they grow slowly.

In a more general sense, it has been shown that the transfer of information inside multi-cellular structures can provide considerable insights into the physiology of these structures. More complex informatics theory may well provide more profound insights into biology. We are progressing to test this idea by calculating theoretical information flows, and mutual information, along the chains of cells using information theory first expounded by Claude Shannon.

Funding Statement

This work was supported by the Defense Advanced Research Projects Agency RadioBio program (USA) [DARPA’s Radiobio program].

Acknowledgments

This work was supported in part by the financial support of DARPA’s RadioBio program. Dr Andrew Eckford, of York University, Toronto, works with the authors on RadioBio, and we would like to thank him for his insights and contributions to this paper.