Time-Fractal in Living Objects

Homeostasis creates self-organized synchrony of the body’s reactions, and despite the energetically open system with intensive external and internal interactions, it is robustly stable. Importantly the self-organized system has scal-ing behaviors in its allometry, internal structures, and dynamic processes. The system works stochastically. Deterministic reductionism has validity only by the great average of the probabilistic processes. The system’s dynamics have a characteristic distribution of signals, which may be characterized by their frequency distribution, creating a particular “noise” 1/f of the power density. The stochastic processes produce resonances pumped by various noise spectra. The chemical processes are mostly driven by enzymatic processes, which also have noise-dependent resonant optimizing. The resonance frequencies are as many as many enzymatic reactions exist in the target.


Introduction
All parts of the biosystems are energetically open. The micro and macro environment have a decisional influence on their processes. The system exchanges energy and information with its environment. According to a well-defined balance, the processes are dynamic and interconnected with each other, the homeostasis [1]. This dynamic stability is self-organized [2], and despite the intensive interactions, it is robustly stable at large order of magnitudes [3]. The dynamic stability is regulated and controlled by the homeostatic feedback mechanisms [4], keeping the balance between promoters and suppressors in the complete system [5]. The living network is undoubtedly not a simple addition of its parts [6]. It forms a complex structure [7]. Theoretical biology faces a severe How to cite this paper: Szasz where X and Y are the signals of the system and outside environment, respectively. Due to the short time realized microstates, the number of diagnostic states is significantly less than the numberof its determinant signals i D , consequently, the microstates appear as statistical statements. The same homeostatic macrostate has a wide variety of microstates that change rapidly over time, fluctuating around the averages. The probability that the microstate falls in the interval ( ) , d + X X X at time t, i.e., the probability density ( ) Consequently i D is given by ( ) it is a stochastic determinant which primarily we characterize with its average (mean value) and its variance The living, dynamic equilibrium is well-regulated but in a probabilistic way. The time-dependent processes realize the observed signal with a probability, as the actual exposition from the possibilities of the fluctuations of the measured signal.
The vital principle is the feedback mechanism, which controls the balance within a predetermined range around the reference value. It is usually well modeled with fuzzy logic, an approach to counting "degrees of truth" rather than the usual "true or false" decisions [35]. This logic governs homeostatic equilibria in all ranges of space and time in living systems. This uncertain value is undoubtedly in a controlled reference interval, were strongly interconnected negative feedback loops regulate the balance in the micro and macro ranges, forming the system's dynamic stability.
These phenomena request a stochastic approach (probability of events dependent on time) instead of conventional thinking based on deterministic changes [36]. Deterministic reductionism can mislead the research. The homeostasis is often ignored and used as a static framework for effects [37]. The stochastic approach is fundamental in biological dynamism [38]. The dynamic homeostatic equilibrium keeps the system in a stable but constantly changing state.

Stochastic and Deterministic Approach
A model calculation of tumor growth shows the strength of the stochastic approach. In a simple example, the growth of a tumor can be described deterministically. The deterministic change of tumor mass ( where k is a constant. A well-known exponential solution uses the mass of the tumor at the start of its observation ( 0 M ): In a deterministic way, the prognostic task of oncology would be simple regarding exponential growth. However, the process is stochastic, requesting the It depends on the added cells to the tumor from the previous time interval ). In a differential equation form: When we start from a single cell ( ( ) Compare (7) and (10)

The Fluctuation Phenomena
The signals follow the living, dynamic interactions, the molecular changes, and The (12) with the Parseval's formula may be evaluated where ( ) S f is the spectral power density in any random stationary case. The Fourier transform of ( ) x t stochastic process is the primary step to study the phenomena [39], A The even function of the frequency, i.e., ( ) ( ) The ( ) S f gives the intensity of noise as a function of spatial frequency, measured in W J Hz = , characterizing the stochastic signal with the f frequency.
The most straightforward complex noise follows normal (Gaussian) distribution (the amplitudes have normal distribution), and its power function ( ) S f is self-similar through many orders of magnitudes. In this simple case, the ( ) The α exponent in (16) formally refers to optics, noted as the "color" of the noise. The white-noise is flat ( 0 α = ), the pink-noise has 1 α = , and other colors are described by various other numbers up to 2 α = , the brown-noise. So, the ( ) S f of pink-noise inversely depends on f frequency, noted as 1/f noise.
The 1/f noise carries the self-similar structure of living processes having a time-fractal covering the life's dynamism [40] [41]. The dynamical fractal structure of living systems marks the self-organizing both in geometric and time structures and dynamically regulates the living matter [42], defines time-fractal structure in stochastic way of the living systems [43], a 1/f fluctuation. The physiological control shows 1/f spectrum [44]. One of the most studied such spectra is the heart rate variability (HRV).
This 1/f noise has a particular behavior. Each octave interval (halving or doubling in frequency) carries an equal amount of noise energy. The living system makes special signal processing due to its self-organized symmetry, so it transforms the white noise to pink [45], forming the most common signal in biological systems [46].

White Noise
All frequencies in the entire interval have the same A amplitude in the white  (17) i.e., from (16), 0 α = . Consequently, the autocorrelation function is completely uncorrelated: The band constraint in a limited interval, up to max f upper-frequency limit affects a longer-term correlation: For example, the completely flat ( ) S f limited to the frequency-band [−10 -10] has well-defined autocorrelation Figure 1: The correlation function oscillates, so the correlation length does not monotonically decrease in band-limited white noise.

The 1/f Noise
According to (14), the Fourier transform of ( ) The Fourier transform of the function ( ) where a is an arbitrary complex number, and f is the frequency: Use (21) and (22) we get: Using Parseval's formula and (15): The living processes are basically self-similar, so it is convenient to define the self-similarity of a stochastic process. A stochastic process is said to be self-similar if the effective power of the stochastic process representation ( ) ( ) ( ) ( ) And so from (22) and (20), we get ( ) Also, for the power spectral density function, the functional equation may be expressed: for every positive scalar a and every scalar f. To solve this equation, we assume that 0 f > and set for a the value a f = : Let us set for a the value a f = and take into account that the power density function is even, so we obtain the 1/f spectrum, or "pink-noise": The autocorrelation function of ( ) 1 1 S f f ∝ pink noise with Fourier transformation has a singular result: Ci ∞ = , the autocorrelation of 1/f noise in long time-lag is zero By the ergodic hypothesis [47], the autocorrelation function of a stationary random process ( ) x t can be defined as where τ is the time-lag. The relation between autocorrelation function and the power density spectrum can be expressed by the Fourier transform of the autocorrelation function (Wiener-Khinchine theorem), namely: From these (considering [36] and [48]), we may conclude Assuming the lower cutoff frequency min f , the function of such an approx- The procedure is also shown in Figure 2.
It can be seen from the figure that here too, there is a problem with the introduction of the correlation length since the correlation function oscillates.
In the case where the lower cutoff frequency is minimal, the argument of the Ci-function is small even at significant offset times. Then the correlation function is as shown in Figure 3.
It appears that this case can be approximated by the sum of white noise and a virtually constant correlation function. More precisely, the can be asymptotically approximated by The autocorrelation function of 1 f α ( (38) Note that colored noises do not fit the white and pink noises, so the basic noises have no common expression.
The pink noise cannot be described with the classical apparatus of nonequilibrium thermodynamics. Macroscopic fluctuation characterizes the Let be an extensive one whose relaxation time is much longer than the others.
Then the fluctuation can be described by this single extensive one. When (39) Solving (41): Then the correlation function is: and its power spectrum: so it follows that The correlation function is constant in this case, so the pink noise correlated in the same way for each shift, so there can be no thermodynamic fluctuation! Starting with such randomized deterministic fluctuations, we get equivalents to form of (40), like: In this case, instead of (41), we get the following spectrum: Assuming that the temporal correlation length probability density function is lognormal, the resulting noise spectrum is: 1 f α . It is the same as the originally white-noise pumped stochastic case. It is confusing, of course, that this process started from deterministic distribution, but it was overcome by assuming that there is a random series of such deterministic fluctuations.
Two stochastic processes can be considered equivalent if their noise spectrum is the same. Based on this, we introduce a stochastic excitation term ( ) q t to Hence the power spectrum The following choice leads to the desired result: Consequently, if ( )

Orstein-Uhlenbeck Process
The power spectrum of a random series of such deterministic fluctuations differs from the white-noise pumped Langevin solution only in a proportionality factor.
We approach the fluctuation by decomposing it into the sum of quasi-periodic stochastic processes of different statistically independent time scales. The quasi-periodic stochastic processes with different time scales also have different frequency scales. All such component processes are assumed to be statistically similar. Note the increase of a stochastic ( ) Assume that  is a smooth function of the , , d

X t t variables and that
( ) X t is continuous: The approach that the observed noise by the emission of subsequent processchains in statistical mechanics, the Markov process [49] describes the chain reaction, which is used in biology too [50].
Since dt can be chosen to be arbitrarily small, the ( ) where notes the mean, and is the standard deviation. Solving function equations where A and D are smooth functions of X and t, and 0 D > . Considering the normality of (55) and (60): is the unit standard deviation squared normal distribution stochastic process with zero means. Turning to a differential equation, we get the following nonlinear generalized Langevin equation driven by normally distributed white noise: In the Gillespie sense [51], the stochastic process is self-similar, resolved to a sum of statistically independent terms normally distributed within the studied interval. Consider the simplest of the self-similar stochastic processes in (61): where τ is the time constant of the process.
The describes an Ornstein-Uhlenbeck process (OUP), which is stochastic and follows a normal (Gaussian) distribution. The OUP is homogeneous in time. Its homogeneity in time allows the OUP to describe it simply with the stochastic interaction of an energy source and the connected energy-consuming system If the distribution is uniform, that is, if,

Importance of the Self-Similarity
The s τ the time constant of the system in (65) generates the stochastic signal.
The s τ can be considered as the natural time scale of the stochastic process that characterizes the two-point correlation function of the stochastic process. Indeed, the two-point correlation function from (65) shows the degree of correlation decreases exponentially with τ time constant: This feature of s τ is the temporal correlation length.
The complexity of the system involves a  (73) improper integrated, we get the desired result: The scale invariance means that the probability scale is independent, In the case where only self-similarity is required, e.g., as a function of density. The self-similar distribution function is thus the condition a shaped power spectrum: The above considerations can be generalized to a large extent. Open Journal of Biophysics We start from the stochastic process described by the equation, using normally distributed white noise as before in (62). Then the power spectrum will be: If we require only self-similarity, we get from (84) and (59) Due to the physical image, the integral is arranged into a form: The self-similarity is again desired the power spectrum: This result concludes to an important note: the self-similarity is a more fundamental feature of the noise than its 1/f shape. Support this we derive instead of the 1 f α the noise spectrum from thermodynamic fluctuations, [55].

Energy Dissipation
Considering that the quantum theory of the dissipative systems is not adequately worked out, we stay within the range of the classical theory. We suppose that the is the white-noise with zero mean value, infinite dispersion, and normal distribution. Let us decompose the ( ) , i j A X t function into three parts: where the ik c elements form a cyclic matrix. is identical for each cell, and at the same way, we may suppose that i D is constant for each cell. This latter can be justified because each cellis to be found in the same heat conditions. We did not assumed any confinement for the ( ) i f t function. The proposed equation isthe generalization of the model of the coupled damped oscillators, which showed [57] that the stochastic resonance is included in the forms of motion. We are going to examine a case where the social signal has low amplitude; therefore, the nonlinear members can be neglected. Then (91):

Cellular Communication in a Noisy Environment
The effective field strength of thermal noise was first calculated by Weaver and Astumian [58]. The Weaver & Astumian model (W-A model) assumed changes in the field strength result from fluctuations of space charges on both sides of the cellular membrane and further showed a thermal noise limit at low frequencies.
Kaune [59] revisited the W-A model and showed that the field strengths typical of thermal noise converge to zero at low frequencies. Therefore, the W-A model does not describe this region appropriately. However, thermal noise in Kaune's model [19] is assumed to be synchronized (coherent) over the entire cell membrane. This assumption is called the coherence condition. Unfortunately, thermal noise is unlikely to be coherent over a large structure such as a cell. Therefore, the calculation that followed is limited to a highly unlikely special case. Kaune set all noise-generators to be equipotential based on the coherence condition by assuming parallel connectivity and the equivalent electrical circuit. As the coherence condition does not hold in the general case, the equipotential assumption also does not hold in the general case. We generalized the problem and developed a solution [60]. Our results proved when there are only zero-mode currents present. The limit does not exist. However, at non-zero currents, the thermal noise does limit the efficacy of electromagnetic effects in low frequencies.
The zero mode is the action by central symmetry for all individual cells instead of the translation symmetry of the usually applied outside field effects.
The topological construction is an essential factor of the cellular organization, [61], irrespective it is alive or not. The cellular structure, because of some topological reasons, develops preferring special coordination arrangements [62] and could arrange a self-organized collectivity [63] [64]. It was discovered that the division tendency is very low in the cell population, small in number [65]. For the start of a significant cell division, a critical cell density is necessary. This was later observed on a self-synchronization of chemical oscillators [66]. The topological importance was assumed in living cellular cultures also, [67], declaring that not the cell density but the position (coordination number) of cells related to each other determines what is favorable or not favorable from the point of view of division. This hypothesis was later justified experimentally [68]. The cells in developed multicellular living objects are grouped into organs to perform certain tasks in a network together. This network extends inside the cells and has suitable connection points outside the cell wall, ensuring with this to involve the cellular mechanisms in the tasks of the network. The cytoskeleton of the cells provides the basic cellular information-transfers intracellularly. The internal cytoskeleton network has transmembrane bridges (e.g., adherent connections, junctions) connecting the matrix structure on the outer side of the cell through the polar protein molecules [69]. The network develops by polymerization [70], where the water structures of aqueous electrolyte arrange the extracellular matrix partially. For example, the formed "intercellular filaments" in epithelial tissues implements the mechanical coupling of individual cells [71] [72]. Ordered water creates efficient proton conduction mechanisms [73] that disordered water does not have. The hydrogen bridges transport the protons, which is crucial in living systems [74]. This high-speed and low dissipation of the transport propagation is based on Grotthuss-mechanism [75].
The healthy cells are under the control of others in the network ("social" signaling [76], a collective action). Social information should spread within the body without loss of information. However, the environment is noisy, and the living information exchange faces this challenge. Now, we are going to prove that among the modes belonging to the eigenvectors of the matrix (93) of equation (91), there are modes of zero noise spectrum. It is well known that any cyclic matrix can be diagonalized by the transformation matrix [77], that is   In consequence, every non-zero order mode is noiseless because: So the zero-order noises are not only limitless by thermal noises, but the signal exchange in such a way is noiseless.

Conclusion
The stochastic processes drive the homeostatic harmony, synchronizes the processes by environmental noises, while the system performs the important internal signal communications noiselessly. The dynamic stochastic living systems involve characteristic resonances. Particular resonant frequencies differentiate and describe the various enzymatic processes.