**Journal Name:** Scholar Journal of Applied Sciences and Research

**Article Type:** Research

**Received date:** 28 May 2018

**Accepted date:** 02 August 2018

**Published date:** 14 August 2018

**Citation:** Naumov MM (2018) Oscillation Energy of Plant
Biological Time in Ontogenesis. Sch J Appl Sci Res. Vol:
1, Issu: 5(44-51).

**Copyright:** © 2018 Naumov MM. This is an open-access
article distributed under the terms of the Creative Commons
Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the
original author and source are credited.

# Abstract

**Background:** Research objective - to calculate energy of oscillation
of biological time of plant in ontogenesis under the influence of weather
environment. At the same time, the simplest model of system for the first
calculation is taken.

**Methods and Findings:** Biological time is represented in the form
of the dependence on photosynthesis and respiration processes of the
whole plant organism via СО2 exchange. Two-dimensional temporal
space is implemented naturally. Consideration is given to the issue of
biological time oscillations during plant ontogenesis. It is believed that
lowering the air temperature from the optimum level down to the actual
temperature would deviate the biological temporal vector from the
physical time axis. The time change is deemed to be an integral of the
scalar product of the biological temporal vector with the physical time
axis. The flow rate of biological time is determined by the rate of CO2
exchange for the whole plant depending on the state of environmental
factors (photosynthetic active radiation (PAR), air temperature, and
productive soil moisture). Oscillations of the newly obtained normalized
logistic curve of plant growth are assigned to those of the whole plant
biological time.

**Conclusions:** The potential and kinetic oscillation energy of
biological time of the whole sunflower plant during ontogenesis is
calculated subject to the condition of environmental factors. The plant
organism is considered as a simplified system or a conservative system.

# Keywords

Development, Energy, Growth, Photosynthesis, Respiration, Time.

# Abstract

**Background:** Research objective - to calculate energy of oscillation
of biological time of plant in ontogenesis under the influence of weather
environment. At the same time, the simplest model of system for the first
calculation is taken.

**Methods and Findings:** Biological time is represented in the form
of the dependence on photosynthesis and respiration processes of the
whole plant organism via СО2 exchange. Two-dimensional temporal
space is implemented naturally. Consideration is given to the issue of
biological time oscillations during plant ontogenesis. It is believed that
lowering the air temperature from the optimum level down to the actual
temperature would deviate the biological temporal vector from the
physical time axis. The time change is deemed to be an integral of the
scalar product of the biological temporal vector with the physical time
axis. The flow rate of biological time is determined by the rate of CO2
exchange for the whole plant depending on the state of environmental
factors (photosynthetic active radiation (PAR), air temperature, and
productive soil moisture). Oscillations of the newly obtained normalized
logistic curve of plant growth are assigned to those of the whole plant
biological time.

**Conclusions:** The potential and kinetic oscillation energy of
biological time of the whole sunflower plant during ontogenesis is
calculated subject to the condition of environmental factors. The plant
organism is considered as a simplified system or a conservative system.

# Keywords

Development, Energy, Growth, Photosynthesis, Respiration, Time.

# Introduction

We cannot see time, even with the strongest microscope. We can only study events and processes and compare time intervals to them. In this paper, we will study time with respect to the photosynthesis and respiration processes of plants, the plant growth processes, and the total dry plant biomass value.

It is known that the duration of plant ontogenesis is not a constant value relative to the physical (calendar) time axis. Thus, for example, the duration of sunflower ontogenesis in the south of Ukraine varies from 80 to 120 days. That is, biological time (simply, the time) tends to expand and compress depending on environmental factors [1-4].

The mathematical apparatus has been described in papers [5-7]. Let us briefly recall the principal points of the theory of oscillations. To begin with, we will consider the simplest oscillatory system described by the equation:

$$\frac{{d}^{2}x}{d{t}^{2}}=f(x)\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\left(1.1\right)$$

It is assumed that function f (x) is integrable and, generally, a nonlinear function of the x-coordinate. We will introduce a new variable: $y=\frac{dx}{dt}$ which allows us to exclude the time in an explicit form from the equations of motion, although, as before: x = x (t) and y = y (t). We can then state:

$$\frac{{d}^{2}x}{d{t}^{2}}=\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}=\frac{dy}{dx}y\text{\hspace{1em}}\text{\hspace{1em}}\left(1.2\right)$$

In new coordinates, equation (1.1) takes the following form:

$$\frac{dy}{dx}=\frac{f(x)}{y}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\left(1.3\right)$$

Having integrated equation (1.3), we obtain [5,7]:

$$\frac{1}{2}{y}^{2}-{\displaystyle \int f(x)dx}=h=const\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\left(1.4\right)$$

This is an expression for two types of energy. As is customary in physics, it speaks of kinetic energy and potential energy.

# Materials and Methods

Let us consider the sum of the effective temperatures method (a method of degree-days). This method is based on the hypothesis that the sum of the effective temperatures necessary for a particular inter-phase development period to occur is constant. This hypothesis is expressed by the equation [4]:

$$n=\frac{A}{{t}_{sr}^{o}-B}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\left(2.1\right)$$

where A - is a constant equal to the sum of the effective
temperatures over the inter-phase development period,
°С; t^{o}_{
sr} is the average air temperature for the inter-phase
development period, in °С; В is the lower limit of temperature
for plant development, in °С; and n is the number of days in
the inter-phase development period.

The moment of the plant development phase (the developmental phases for sunflowers are shoots, the second couple of real leaves, budding, blossoming, and full maturity of seed) is calculated using two equations. First, the effective temperature of the current day of calculation is to be computed:

$$\Delta {t}_{d}^{o}={t}_{d}^{o}-B\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\left(2.2\right)$$

where t^{о}d is the average air temperature of the current
day of calculation, in °С, and Δt_{o}d is the effective temperature
of the current day of calculation, in °С. Then, cumulatively,
the sum of the effective temperatures for each day j of the
inter-phase development period n is to be calculated:

$${t}_{s}^{o}={\displaystyle \sum _{j=1}^{n}\Delta {t}_{d}^{oj}}\text{\hspace{1em}}0\text{}\le {t}^{o}{}_{s}\le \text{}A\text{\hspace{1em}}\text{\hspace{1em}}\left(2.3\right)$$

where j is the number of days of the development period
and t^{o}s is the sum of the effective temperatures for the interphase
development period, in °С. Thus, the time under this
method is measured in Celsius degrees instead of in true
time units. To move toward real time, we will perform
the following operation: by the time the sum of effective
temperatures for the inter-phase development period
tos has passed (this can also be one from “shoots” to “at
full maturity of seed”), equal to A, it is considered that the development phase has occurred. We divide the right and
the left sides of equation (2.3) by the sum of the effective
temperatures equal to A, in °С. This means that we have
normalized the sum of the effective temperatures. Now this
sum is expressed by an analogue, which varies from 0 to 1 of
the real time. The effective temperature of the current day of
calculation, Δt^{o}_{
d}, will also be expressed in relative units. This
value corresponds to one day of physical (calendar) time.
Thus, we have moved from the sum of effective temperatures
method to the normalized real physical time. During the
entire ontogenesis period, the time varies from some small t_{0}
up to 1 (from the “shoots” event to “at full maturity of seed”).
The normalized physical time also depends on the average
air temperature for the current day of calculation (equation
(2.2)). Let us consider the elementary increment of the
normalized real physical time for one day of calculation
(2.2) under the sum of effective temperatures method.
This increment of physical time depends on the average air
temperature for the current day of calculation. Thus, if the
air temperature is optimal (for example, 25 °С average air
temperature of the settlement period for the sunflower),
then the increment of the normalized physical time will
be the largest, maximum ΔT_{max}= max. If the average air
temperature for the current day of calculation dropped to a
biological minimum temperature В = 10 °С (for sunflowers),
then the increment of the normalized real physical time
would be zero. That is, we can imagine that the temporal
vector deviates from its normal position, the physical time
axis. Such a physical time axis has a direction - from the past
to the future. Then, the projection of this deviating vector
onto the normal physical time axis will give a real time
increment ΔT_{real}, Figure 1.

**Figure 1:** Deviation of the maximum temporal vector ΔT_{max} under the sum
of effective temperatures method from the normal physical time axis when
the average daily air temperature for the current day of calculation has
lowered from the optimal average daily air temperature down to the real one.

The ΔT_{max}value represents the maximum increment of
the temporal vector at the optimal average air temperature
for the current day of calculation. When the average air
temperature for the current day of calculation lowers,
the temporal vector deviates. Thus, at the temperature
of biological zero, or В temperature (equation (2.2)), the temporal vector ΔT_{max} will be orthogonal to the normal
physical time axis, and the deviation angle α will equal π / 2,
then the projected ΔT_{max} of the deviating temporal vector onto the normal
physical time axis will be zero: ΔT_{real}= 0 (cos π/2 = 0), and
development of the plant (sunflower) will be suspended.
Thus, the angle α of deviation of the real temporal vector
depends on the average air temperature of the current day
of calculation. That is, with equation (2.2), we calculate the
scalar product of the deviation of the temporal vector ΔT_{max}
from the physical time axis. From this, it can be seen that
with the sum of effective temperatures method, we can
calculate the sum of the scalar products of the deviation
of the real temporal vector ΔT_{max} from the physical time
axis. Moving to the limit when the increment ΔTmax tends
toward zero, we obtain the integral of the scalar product
(Budak B.M., Fomin S.V. 1967). In this case, the line (2.3)
turns into a curve in some two-dimensional temporal space,
Figure 1, (Naumov M.M. 2004, 2005) [8.9]. Thus, we obtain a
curvilinear integral:

$$T={\displaystyle \underset{{\tau}_{b},{\tau}_{e}}{\int}(F,d\tau )}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\left(2.4\right)$$

where T is the value of the real normalized biological
time, in relative units; (F, dτ) is the scalar product of the
deviation of the real biological temporal vector from the
physical time axis; dτ is the vector ΔT_{max} when it tends
toward infinitesimal; F is the deviating vector ΔT_{max} at the
angle α when the average daily air temperature lowers
from the optimal temperature to the real one; and τb and τe
are the starting point (“shoots”) and the end point (“at full
maturity of seed”) of the biological time curve, respectively.

$$T(t)={\displaystyle {\int}_{{t}_{b}}^{{t}_{e}}\left[P(t)\frac{d\varphi (t)}{dt}+Q(t)\frac{d\psi (t)}{dt}\right]}dt,\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1.\text{\hspace{1em}}\text{\hspace{1em}}\left(2.5\right)$$

Here, P (t) and Q (t) are some functions to be defined
later on and t is the physical time axis, in relative units,
where for the entire ontogenesis (from the “shoots” event
to the “at full maturity of seed” event), the time t varies from
some small t_{0} (corresponding to the “shoots” event: let i be,
for example, t_{0} = 0.01) to 1 (“at full maturity of seed” event).
Let us differentiate this already calculated integral (2.5) with
respect to t. We obtain:

$$\frac{dT(t)}{dt}=P(t)\frac{d\varphi (t)}{dt}+Q(t)\frac{d\psi (t)}{dt},\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1.\text{\hspace{1em}}\text{\hspace{1em}}\left(2.6\right)$$

Now we write down the equation (Davidson J.L., Philip
J.R. 1958), which shows that the incremental change of the
dry biomass of the whole plant depends on its CO_{2} exchange:

$$\frac{dM(t)}{dt}=\frac{d\u0424(t)}{dt}-\frac{dR(t)}{dt},\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1.\text{\hspace{1em}}\text{\hspace{1em}}\left(2.7\right)$$

where M (t) is the total dry plant biomass value, g ∙
plant-1; $\frac{d\u0424(t)}{dt}$
is the photosynthesis rate of the whole plant
in ontogenesis, g ∙plant^{-1} ∙day^{-1}; and $\frac{dR(t)}{dt}$
is the respiration
rate of the whole plant in ontogenesis, g ∙ plant^{-1}∙day^{-1}. Let us
write down the equation (2.7) in another form:

$$dM(t)=\left(\frac{d\u0424(t)}{dt}-\frac{dR(t)}{dt}\right)dt,\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1,\text{\hspace{1em}}\text{\hspace{1em}}\left(2.8\right)$$

and represent the total dry plant biomass value M (t)
in the normalized form. It is known that accumulation of
the total dry biomass of the whole plant is subject to the
S-shaped growth pattern [10]. Now we take the value of
the final total dry biomass of the whole plant and divide it
by the current value of the total dry plat biomass. Then, we
obtain that the total dry biomass of the whole plant will vary
in relative units: from small M_{0} corresponding to the shoots
event to 1 corresponding the at full maturity of seed event.
The S-shaped growth will be maintained and at the end of
ontogenesis, for the “at full maturity of seed” event, the total
dry plant biomass will equal μ = 1. Now we replace M (t) with
the normalized values of the total dry plant biomass μ (t) in
equation (2.8). We obtain:

$$d\mu (t)=\left(\frac{d{\u0424}_{\mu}(t)}{dt}-\frac{d{R}_{\mu}(t)}{dt}\right)dt,\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1,\text{\hspace{1em}}\text{\hspace{1em}}\left(2.9\right)$$

where Ф_{μ}(t) is the photosynthesis output value of the
whole plant during ontogenesis, which depends on the
physical time t, in relative units and Rμ(t) is the respiration
output value of the whole plant during ontogenesis, which
depends on the physical time t, in relative units. We write
down equation (2.6) in the same form and compare it with
equation (2.9):

$$dT(t)=\left(P(t)\frac{d\varphi (t)}{dt}+Q(t)\frac{d\psi (t)}{dt}\right)dt,\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1,\text{\hspace{1em}}\left(2.10\right)$$

Plant age is determined by the growth and development of the entire organism. That is, plant age is determined by the total dry biomass value, comprising the development of its individual organs and systems (leaves, roots, stems, reproductive organs, conducting systems, xylems, phloem, etc.). Thus, if, for example, the value of the normalized total dry biomass of the whole plant is μ (t) = 0.3, then the real biological time of ontogenesis is equal to 0.3 relative units of biological time, too. In this case, we will assume that the biological time corresponds exactly to the normalized value of the total dry plant biomass μ (t). From this, we obtain that the functions in equation (2.10) match exactly the photosynthesis and respiration rates of the whole plant:

$$P(t)\frac{d\varphi (t)}{dt}=\frac{d{\u0424}_{\mu}(t)}{dt},\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1,\text{\hspace{1em}}\left(2.11\right)$$

$$Q(t)\frac{d\psi (t)}{dt}=-\frac{d{R}_{\mu}(t)}{dt},\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1.\text{\hspace{1em}}\left(2.12\right)$$

Hence, change of biological time is entirely dependable
on the processes of photosynthesis and respiration of the
plant, meaning on the СО_{2} exchange of the whole organism.
Because the photosynthesis and respiration rates depend
on the culture genotype and environmental factors
(temperature, productive soil moisture, light, mineral
nutrition and other factors), then the biological time will
depend on these factors, too:

$$\frac{dT(t)}{dt}=\frac{d{\u0424}_{\mu}(t)}{dt}-\frac{d{R}_{\mu}(t)}{dt},\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1.\text{\hspace{1em}}\left(2.13\right)$$

Now let us consider equation (2.9) which shows the
incremental change of the total dry biomass in normalized
form with respect to the normalized time, where during the
entire ontogenesis the normalized physical time t varies
from some small t_{0} to 1:

$$\mu (t)={\displaystyle \underset{{t}_{0}}{\overset{1}{\int}}\left(\frac{d{\u0424}_{\mu}(t)}{dt}-\frac{d{R}_{\mu}(t)}{dt}\right)dt}={\displaystyle \underset{{t}_{0}}{\overset{1}{\int}}\frac{d\mu (t)}{dt}dt}={\displaystyle \underset{{t}_{0}}{\overset{1}{\int}}d\mu (t)}\text{\hspace{1em}}\left(2.14\right)$$

Here, the total СО2 exchange in the plant determines the
increments and growth of the normalized total dry biomass,
where t_{0} is the moment of growth beginning, and 1 is the
moment of growth completion, that is, from the shoots event
to the at full maturity of seed event.

For further calculation of the total kinetic and potential energy of the whole plant organism, we need a logistic growth curve for the total dry plant biomass in normalized form. We have also obtained that it is removed by means of a theory of oscillation, as verified in preceding papers (Naumov M.M. 2004, 2005, 2010, 2011) [8,9,11,12] which has the following form:

$$\mu (t)=-\frac{1}{{\omega}_{0}}\mathrm{sin}({\omega}_{0}t+{\beta}_{0})+t,\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1,\text{\hspace{1em}}\left(2.15\right)$$

where ω_{0} is the cyclic oscillation frequency equal to ${\omega}_{0}=\frac{2\pi}{{T}_{0}}$
[5], in which T0 is an oscillation period equal to the whole
ontogenesis, starting from the shoots event until the at full
maturity of seed event, T_{0} = 1; and β_{0} is the initial phase of
development of oscillations of the normalized physical plant
time (let us assume that for the growth state, for the shoots
event, the initial phase of development of oscillations is equal
to β_{0}= 0.01 radian). The first derivative of the normalized
logistic growth curve of the total dry biomass of the whole
plant with respect to the physical time t is as follows:

$$\frac{d\mu (t)}{dt}=-\mathrm{cos}({\omega}_{0}t+{\beta}_{0})+1,\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1,\text{\hspace{1em}}\left(2.16\right)$$

This equation shows the change in the growth rate of
the normalized total dry biomass of the whole plant and
depends entirely on the current state of the environmental
factors such as PAR, heat, productive soil moisture, nutrition,
СО_{2} content in the air and other unconsidered factors. It
should be mentioned that some of the factors influencing the
growth processes of the normalized total dry biomass of the
whole plant in equation (2.16) have not been considered yet.

The second derivative of the normalized logistic growth curve of the total dry biomass of the whole plant with respect to the physical time t is as follows:

$$\frac{{d}^{2}\mu (t)}{d{t}^{2}}={\omega}_{0}\mathrm{sin}({\omega}_{0}t+{\beta}_{0}),\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1.\text{\hspace{1em}}\left(2.17\right)$$

Here, as well, we have not considered the effects of environmental factors on the second derivative of the normalized growth rate of the logistic curve.

Thus, taking into account equations (2.9), (2.13), and (2.14), the biological time oscillations T in plant ontogenesis, having excluded the effect of the environmental factors, would be presented as follows:

$$T(t)=-\frac{1}{{\omega}_{0}}\mathrm{sin}({\omega}_{0}t+{\beta}_{0})+t,\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1,\text{\hspace{1em}}\left(2.18\right)$$

$$\frac{dT(t)}{dt}=-\mathrm{cos}({\omega}_{0}t+{\beta}_{0})+1,\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1,\text{\hspace{1em}}\left(2.19\right)$$

$$\frac{{d}^{2}T(t)}{d{t}^{2}}={\omega}_{0}\mathrm{sin}({\omega}_{0}t+{\beta}_{0}),\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1.\text{\hspace{1em}}\left(2.20\right)$$

Thus, equations (2.18) (2.19), and (2.20) describe
the biological time oscillations of the whole organism,
regardless of the current condition of environmental
factors in the normalized form. These equations describe
the genetic features of plants in relative time units, where
during the entire ontogenesis, the physical time t varies
from some small value (for the shoots event) t_{0} to 1 (for the
at full maturity of seed event).

Now we can compute the kinetic and potential energy of biological time oscillations T in plant ontogenesis in relative units with equation (1.4).

The kinetic energy K(t) of the biological time oscillations of the whole plant organism, regardless of the environmental factors in the normalized form, is as follows:

$$K(t)=\frac{1}{2}{y}^{2}=\frac{1}{2}{\left(-\mathrm{cos}({\omega}_{0}t+{\beta}_{0})+1\right)}^{2},\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1.\text{\hspace{1em}}\left(2.21\right)$$

Having the brackets removed and similar terms collected, we obtain the following equation for the kinetic energy of the biological time T oscillations of the whole plant, regardless of the current condition of environmental factors:

$$K(t)=\frac{1}{4}\mathrm{cos}\left(2{\omega}_{0}t+2{\beta}_{0}\right)-\mathrm{cos}\left({\omega}_{0}t+{\beta}_{0}\right)+\frac{3}{4},\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1.\text{\hspace{1em}}\left(2.22\right)$$

The equation of the kinetic energy of biological time oscillations of the whole plant (2.22) has three terms: the first two are dynamic, ontogenesis time-dependent and responsible for addition of two harmonic oscillations of biological time in ontogenesis, and the third one is a constant responsible for positive values of kinetic energy. The constant of kinetic energy i should say that the vital processes permanently occur.

The potential energy V(t) of the biological time oscillations of the whole plant, regardless of the environmental factors in the normalized form, is as follows:

$$V(t)=-{\displaystyle \int f(x)dx=}-{\displaystyle \underset{}{\overset{}{\int}}{\omega}_{0}\mathrm{sin}\left({\omega}_{0}t+{\beta}_{0}\right)\cdot \left(-\mathrm{cos}({\omega}_{0}t+{\beta}_{0})+1\right)dt},\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1.\text{\hspace{1em}}\left(2.23\right)$$

We have recorded the potential energy directly for the physical time variable t, as equation (2.19):

$$dT(t)=\left(-\mathrm{cos}({\omega}_{0}t+{\beta}_{0})+1\right)dt$$

By integrating this equation, we obtain a dependence of the change in the potential energy of the biological time oscillations of the whole plant in ontogenesis, regardless of the current condition of environmental factors:

$$V(t)=-\frac{1}{4}\mathrm{cos}\left(2{\omega}_{0}t+2{\beta}_{0}\right)+\mathrm{cos}\left({\omega}_{0}t+{\beta}_{0}\right)+{C}_{V},\text{\hspace{1em}}{t}_{0}\le \text{}t\text{}\le \text{}1.\text{\hspace{1em}}\left(2.24\right)$$

where C_{V} is the constant of integration, a constant of
potential energy of the biological time oscillations. The
constant value of potential energy CV shall be such that the
potential energy V (t) is always positive and equal to zero
in the minimum point. Such a constant was found by the
potential energy curve fitting method with calculations on PC in the MATLAB system. Then, the constant of integration
of the potential energy of the biological time oscillations
of the whole plant, regardless of the current condition of
environmental factors is C_{V}= 5/4. Now, we can figure the
kinetic and potential energy of the biological time oscillations
of the whole plant, regardless of the current condition of
environmental factors in relative units during ontogenesis,
Figure 2.

**Figure 2:** Change in the kinetic K (t), potential V (t) and the total energy h =
const of the biological time oscillations of the whole plant organism during
ontogenesis, in relative units.

As seen at Figure 2, the kinetic energy of the biological time oscillations of the whole organism at the beginning and the end of growth takes a zero value. Halfway through the plant growth, at the middle point of ontogenesis t = 0.5, the kinetic energy of the biological time oscillations takes the maximum value K (t) = max. The potential energy V (t) at the beginning and at the end of growth has a maximum value. The potential energy of the biological time oscillations at the end of ontogenesis, at the at full maturity of seed event, is fully stored in the reproductive organs, the plant seeds (in sunflower seeds). Similarly, we can note that the potential energy of the biological time oscillations at the beginning of growth, at the sowing event, is entirely contained in the plant (sunflower) seeds, and for the shoots event, it has some lower value, as potential energy has been spent on the growth and development of the plant organism.

The total energy of the biological time oscillations of the whole plant in accordance with the energy conservation law for a conservative system always remains constant: h = K (t) + V (t) = const. Now let us examine the change in the kinetic and potential energy of the biological time oscillations of the whole plant under real ontogenesis. To do this, we take the sunflower that used to grow in the real conditions of the Odessa region, Chernomorka village in 1986.

Example of calculating the potential and kinetic energy of biological time oscillations of the whole sunflower plant in a real environment in 1986.

For the calculation, we will follow the following pattern.
First, we calculate the tension level of environmental factors
(PAR, heat, productive soil moisture) for every day of
ontogenesis. To do this, we take the average decade data and
assume that during the given calculation decade of growth
and development of the sunflower, the values of PAR,
average daily air temperature, and productive moisture
reserves within a 1-meter-deep soil layer were at the same
level. We assume that the level of soil nutrition was optimal.
The CO_{2} content in the air corresponds to the normal level.

Then, we calculate the change in the normalized physical time t with respect to the calendar time j. Next, we calculate the growth rate of the total dry biomass of the whole sunflower plant in real environmental conditions, taking into account the current environmental factors. Then, we calculate the real dynamics of the total dry sunflower plant biomass during ontogenesis and compare it with the measured values of the total dry biomass in 1986. Further, we will require the final calculated value of the total dry sunflower plant biomass. Then, we will normalize the growth rate of the total dry sunflower plant biomass to obtain the normalized values of the plant biological time T. Finally, we calculate the kinetic and potential energy of the biological time oscillations of the sunflower plant under the real environmental conditions of 1986.

Let us calculate the tension level of environmental factors.
To integrate, we need to choose the integration interval. The
integration interval for us cannot be less than one day. It
is known that there are daily dynamics of photosynthesis
and respiration of plants, and these photosynthesis and
respiration dynamics are not considered in our calculations
for the time being. We will compute the average change in
photosynthesis and respiration for the given calculated
day. Therefore, for our calculations, we have chosen the
integration interval to be one day. The level of tension of
the environmental factors U_{ee}(j) is thus calculated by the
equations:

Light (Gulyaev BI, 1983) [13]

$${\chi}_{light}(j)=1-\mathrm{exp}\left(-{C}_{i}\cdot {I}_{opt}\cdot {I}_{S}(j)\right)\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\left(3.1\right)$$

Heat (Horie T, 1977) [14]

$${\lambda}_{warmly}(j)=1-{a}_{t}{({t}_{opt})}^{2}{\left({t}_{r}^{o}(j)-1\right)}^{2}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\left(3.2\right)$$

Soil moisture (Polevoy AN, 1983) [15]

$${\gamma}_{moisture}(j)=1-{a}_{w}{({W}_{opt})}^{2}{\left({W}_{S}(j)-1\right)}^{2}\text{\hspace{1em}}\left(3.3\right)$$

where j is the day number in ontogenesis (from the
shoots event to the at full maturity of seed event), in calendar
time; χ_{light}(j) is the exponential normalized light curve for the
photosynthesis rate of sunflower culture, in relative units;
I_{opt} is the optimal PAR flow at the upper sunflower sowing
limit for the growth and photosynthesis processes to run
at maximum rate, I_{opt}= 650 W·m^{-2}; I_{S}(j) is the normalized
real PAR flow at the upper sowing limit, in relative units;
C_{i} is the light curve parameter which, along with value I_{opt},
determines the degree of the light curve inflection, Ci=0.01
m^{-2}·W^{-1}; λ_{warmly}(j) is the normalized temperature curve parabola for the growth and photosynthesis of sunflower
culture, in relative units; to
opt is the optimal average daily air
temperature for the growth and photosynthesis processes
of sunflower culture, t^{o}_{
opt}=25 °С; t^{o}_{
r}(j) is the normalized real
average daily air temperature within the sowing period,
in relative units; a_{t} is the temperature curve parameter
which, along with value t_{opt}, determines the critical points
for the growth and photosynthesis processes of sunflower
culture to occur, a_{t} =0.0022 °^{С-2}; γ_{moisture}(j) is the normalized
moisture curve parabola for the growth and photosynthesis
of sunflower culture, in relative units; Wopt is the optimal
reserve of productive moisture within a 1 meter-deep soil
layer for the processes of growth and photosynthesis to
occur, which for the Odessa region, Chernomorka village
is W_{opt} = 131.9 mm; W_{S}(j) is the normalized real reserve of
productive moisture within a 1 meter-deep soil layer, in
relative units; and a_{w} is a parameter of the moisture curve
for the growth and photosynthesis of sunflower culture
which, along with value W_{opt},
determines the critical points
for the photosynthesis process to occur, a_{w} =0.00004 mm^{-2}.

To calculate the PAR flow at the upper sunflower sowing limit equation is used:

$${I}_{PAR}(j)=\left[12.66\cdot S{S}^{1.31}(j)+315{(a+b)}^{2.1}\right]\cdot \frac{41868}{3600{\tau}_{day}(j)}\text{\hspace{1em}}\left(3.4\right)$$

where I_{PAR}(j) is the PAR flow at the upper sowing limit,
W·m^{-2}; SS (j) is the duration of sunshine hours per day, in hrs;
(a + b)=sinh_{0}(j), where: h_{0}(j) is the Sun’s midday height; and
τ_{day}(j) is the length of daylight hours, in hrs.

We only have to compute the normalized values of the PAR flow, the average daily air temperature and the reserves of productive moisture within a 1-meter-deep soil layer:

$${I}_{S}(j)=\frac{{I}_{PAR}(j)}{{I}_{opt}};\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\left(3.5\right)$$

$${t}_{r}^{o}(j)=\frac{{t}_{air}(j)}{{t}_{opt}};\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\left(3.6\right)$$

$${W}_{S}(j)=\frac{{W}_{stocks}(j)}{{W}_{opt}};\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\left(3.7\right)$$

where t_{air}(j) is the current average daily air temperature,
in °С and W_{stocks}(j) is the current reserve of productive
moisture within a 1-meter-deep soil layer, in mm.

Finally, the level of tension of the environmental factors
U_{ee}(j) can be calculated with Liebig-Blackman’s principle,
which considers the impact of each factor on the growth and
photosynthesis rates for sunflower culture for every day of
ontogenesis:

$${U}_{ee}(j)={\chi}_{light}(j)\cdot {\lambda}_{warmly}(j)\cdot {\gamma}_{moisture}(j)\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\left(3.8\right)$$

Now, we should calculate the normalized physical time axis t with respect to the calendar time. Calculation of this normalized time will be based on the formulas [16,17]:

$$t(j+1)={t}_{0}+t(j)+{U}_{ee}(j)\cdot \Delta {t}_{opt},{t}_{0}\le \text{}t\left(j\right)\text{}\le \text{}0.5\text{\hspace{1em}}\text{\hspace{1em}}\left(3.9\right)$$

$$t(j+1)=t(j)+\frac{\Delta {t}_{opt}}{{U}_{ee}(j)},0.5\text{}\le \text{}t\left(j\right)\text{}\le \text{}1.\text{\hspace{1em}}\text{\hspace{1em}}\left(3.10\right)$$

where j is the day number in ontogenesis from the shoots
event to the at full maturity of seed event; t_{0} is the initial
value of the normalized physical time axis (for the shoots
event), in relative units: t_{0}= 0.01; t (j) is the normalized value
of the physical time axis, which depends on the day number
in ontogenesis j, in relative units: t_{0} ≤ t ≤ 1; Δt_{opt} is a constant
related to the maximum development rate for optimal values
of environmental factors, for sunflowers: Δt_{opt}= 0.0097
relative physical normalized time units ∙ day^{-1}. To calculate
the optimal rate of the plant development, Δt_{opt} (sunflower),
it is necessary to take the average long-term value for the
duration of the shoots-to-at full maturity of seed period of
development. For example, let us assume the duration of
this period of development to be 103 days. Then, the value
Δt_{opt} will be calculated from Δt_{opt}= 1/103 = 0.0097 relative
units of the normalized physical time per day. Equations for
calculating the normalized physical time axis t were obtained
and verified in literature [16,17]. According to those studies,
the moment when the relative time axis t (j) is equal to 0.5
corresponds to the blossoming event of the sunflower.

Now, we will calculate the dynamics of the real growth rate of the total dry biomass of the sunflower plant, taking into account the influencing environmental factors (PAR, average daily air temperature, productive soil moisture), because the process of the growth rate of the total dry biomass is affected by current environmental factors, shown in equation (2.8), which shows that the growth rate of the plant is completely dependent on environmental factors through the processes of photosynthesis and respiration of the culture [11,12]:

$${M}_{real}(j+1)={M}_{real}(j)+{U}_{ee}(j)\frac{\Delta {M}_{\mathrm{max}}}{\Delta t}\left(-\mathrm{cos}({\omega}_{0}t(j)+{\beta}_{0})+1\right),\text{}\text{\hspace{1em}}\left(3.11\right)$$

where M_{real}(j) is the real total dry biomass of the whole
sunflower plant, g ∙ plant^{-1} and $\frac{\Delta {M}_{\mathrm{max}}}{\Delta t}$
is the maximum growth
rate of the total dry biomass of the whole sunflower plant
for one day of calendar time j in optimal environmental conditions, here Δt = 1 day, g∙plant^{-1}∙day^{-1}. The maximum
growth rate is observed in the middle of ontogenesis and is
equal to $$\frac{\Delta {M}_{\mathrm{max}}}{\Delta t}$$
=12.1 g∙plant^{-1}∙day^{-1} (this value was obtained
by the author as a result of experimental observations
of sunflower sowing in the village of Chernomorka in
1986 and numerical experiments on a PC in the MATLAB
system). Integrating equation (3.11) for every current day
of calculation, from the shoots event to the at full maturity of
seed event, we obtain the value of the final total dry biomass
of the whole sunflower plant, which amounts to the value
M_{final} = 924.6 g ∙ plant^{-1}, Figure 3.

Now we can normalize equation (3.11) by the value of
the final total dry biomass of the whole sunflower plant
organism M_{final}:

$$\mu (j+1)=\mu (j)+{U}_{ee}(j)\frac{\Delta {M}_{\mathrm{max}}}{\Delta t\cdot {M}_{final}}\left(-\mathrm{cos}({\omega}_{0}t(j)+{\beta}_{0})+1\right).\text{\hspace{1em}}\left(3.12\right)$$

Such normalization will allow calculation of the kinetic energy of the biological time oscillations of the whole sunflower plant in the real growing conditions in 1986 with equation (3.14), see equations (2.21) and (2.22). Let us denote the variable G (j):

$$G(j)={U}_{ee}(j)\frac{\Delta {M}_{\mathrm{max}}}{\Delta t\cdot {M}_{final}}.\text{}\text{\hspace{1em}}\left(3.13\right)$$

Then, the kinetic energy of the biological time oscillations of the whole sunflower plant organism K (j) will equal:

$$K(j)=\frac{1}{4}{\left(G(j)\right)}^{2}\mathrm{cos}\left(2{\omega}_{0}t(j)+2{\beta}_{0}\right)-{\left(G(j)\right)}^{2}\mathrm{cos}\left({\omega}_{0}t(j)+{\beta}_{0}\right)+{\left(G(j)\right)}^{2}\frac{3}{4}.\text{\hspace{1em}}\left(3.14\right)$$

The plot of the obtained kinetic energy of biological time oscillations of the sunflower plant grown in 1986 in the Odessa region, Chernomorka village is presented at Figure 4.

The potential energy of the biological time oscillations of the sunflower plant organism V (j) is calculated from the fact that the total energy is always constant for a conservative system: h = K (t) + V (t) = const and that there is a maximum value for the kinetic energy in ontogenesis (Figure 2). Then, the potential energy of the biological time oscillations of the sunflower plant can be calculated from the fact that we assume the sunflower plant to be a conservative system in the first approximation. This means that the total energy h is always constant, even under the real environmental conditions, although these environmental factors vary very dynamically. Let us consider, however, that h = const (1986). Thus, since at the middle time point of ontogenesis, the value of kinetic energy of the plant biological time oscillations reaches its maximum value, while at the same moment of ontogenesis the potential energy of the plant biological time oscillations equals zero (Figure 2), then we can calculate the potential energy with the equation:

$$V(j)={K}_{\mathrm{max}}(1986)-K(j),\text{\hspace{1em}}\left(3.15\right)$$

where K_{max} (1986) is the maximum value of the kinetic
energy under the growing conditions of sunflower culture
in 1986 in the Chernomorka village, which is K_{max}(1986)
=2.0092∙10^{-4} day^{-2}. The plot of the obtained potential energy
of biological time oscillations of sunflower plant organism
is shown in Figure 4. According to our data on sunflower
growing in 1986, the blossoming event for sunflowers occurred on the 63rd day from the shoots event, which
corresponds to the maximum kinetic energy of the biological
time oscillations of the sunflower plant in ontogenesis. The
same blossoming event for the sunflower corresponds to the
minimum potential energy of the biological time oscillations
of the sunflower plant.

# Discussion

We started this study with the analysis of the sum of
effective temperatures method (degree-days) and came to
the conclusion that the rate of plant development depends
on the processes of photosynthesis and respiration of the
culture, or on the CO_{2} exchange of the whole plant. At the same
time, one can consider abstract intervals of time, not just the
sum of effective temperatures method. If we assume that
the time inside the plant expands and compresses linearly
depending on the environmental factors, we will obtain
the same conclusions based on the fact that the temporal
vector rotates around the point of the current moment of
ontogenesis. We calculated the projection of the deviating
temporal vector onto the physical time axis. As a result, the
biological time was calculated using the integration of the
scalar product of two temporal vectors. This means that we
obtained the meaning of work in the physical sense of the
word, which produces time.

It is known that the rate of plant development is
determined by the factors of heat, productive soil moisture,
light, and plant nutrition. In our case, we have obtained
that the rate of change of the biological time T is entirely
determined by the total CO2 exchange of the culture, which
in turn is determined by the factors of light, heat, productive
soil moisture, nutrition and other unconsidered factors. For
example, changing the content of CO_{2} in the air affects the
rate of development.

It should be noted that the study gave the equation of harmonic oscillations of the biological time T(t) as:

$$\frac{{d}^{2}T(t)}{d{t}^{2}}+{\omega}_{0}^{2}T(t)={\omega}_{0}^{2}t$$

This equation can be regarded as the effect of the driving force of physical time t on the harmonic oscillations of biological time T(t). At the same time, it is known that the processes of photosynthesis and respiration of a plant, first of all, depend on incoming light waves (PAR). Therefore, the equation of the harmonic oscillations of biological time must be matched to the oscillations of light waves.

Also note that this equation of the harmonic oscillations of biological time has been established for the whole organism of the plant. On the other hand, the plant consists of growing organs—leaves, stems, roots, and reproductive organs. The growth of these organs is also described by the obtained logistic growth curve in the normalized form (2.15). Thus, the biological time of the organ will also be described by the equation of the harmonic oscillations of biological time, up to the cells. Studies of isotopic monitoring of cell metabolism show that carbon metabolism in the photosynthetic cell is an oscillatory process [18]. However, this is the material of another study.`

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