Coquaternion Transformations in Nonequilibrium Dynamics of Biologic Systems

Abstract

Imaginary coquaternions cℍ can be represented by matrices of negative feedback $N^{-}$, positive feedback $P^{+}$, and reciprocal links $R^{\pm }$. An added environmental element $E^{\pm }$ endows biologic systems with the structure of cℍ module. Although cℍ representation links base patterns with the geometric structure of the pseudo-Euclidean ${\mathbb{R}}_2^4$ space, unknown physiologic aspects of relationships between base elements may add new functional features to the structure of a functional module. Another question is whether achieving and remaining in the equilibrium state provides stability for a biologic system. Considering the property of a biologic system to return deviated conditions to the equilibrium, the system of ordinary differential equations describing the behavior of a mechanical pendulum was modified and used as a basic tool to find the answers. The results obtained show that in evolving systems, the regulatory patterns are organized in a sequence $NPRN$ of base elements, allowing the system to perform a high amount of energy-consuming functions. In order to keep dissipating energy at the same level, the system bifurcates and finalizes its regulatory cycle in $R^{\pm }$ by splitting $P^{+}$ after which the next cycle may begin. Obtained flows are continuous pathways that do not interfere with equilibrium states, thus providing a homeostasis mechanism with nonequilibrium dynamics. Functional transformations reflect changes in the geometry and metric index of the coquaternion. Related coquaternion dynamics show the transformation of a hyperbolic hyperboloid into the closed surface, which is the fusion of the portions of the hyperbolic hyperboloid and two spheres.

1. Introduction

The notion of a biologic system (BS) has a broad meaning. In this paper, it will be used regarding biologic units, which are morphologically separable and functionally stable as reproducible elements. Thus, the notion of a biologic system can be applied to biological cells, organs, and functional systems, such as cardiovascular, endocrine, gastrointestinal, etc. composed from the previously determined elements [1,2,3,4,5].

At each of these functional (hierarchical) levels, negative feedback $N^{-}$, positive feedback $P^{+}$, and reciprocal links $R^{\pm }$ ($PNR$) have been observed and used as the linear functional patterns formalized from feedback circuits, regulating the functions of two morphologic elements, or subsystems united in two-element biologic systems. Expressed in a matrix form, $PNR$ represent the basis elements $\{S_0,S_1,S_2\}$ of the Lie algebra $sl(2,\mathbb{R})$ of the special linear group $SL(2,\mathbb{R})$ and an imaginary (vector) basis of coquaternions cℍ [3,6,7,8,9,10,11,12] (Figure 1).

Coquaternions and split-quaternions are synonyms. Term coquaternion will be used after Cockle, the author who discovered and named this algebraic structure [13]. If the identity element of a coquaternion is considered an environmental regulatory pattern modulating the functions of biologic systems, the four basis elements of the coquaternion will represent the full set of four base regulatory patterns of a biologic system. The closed algebraic structure of cℍ implies that a BS has a property to be also closed under the physiologic compositions of the base $PNR$ elements. Based on this correspondence, it is assumed that coquaternions represent a functional module of a biologic system as a uniform regulatory structure of biologic cells, organs, and functional systems [14].

Biologic systems are dissipative structures and linear base $PNR$ elements provide the first (rough) approximation of the real biologic processes. Although the 4-space of cℍ is not homogeneous, coordinate basis and formal algebraic operations make basis elements equally responsible for the corresponding functional dynamics and obtained results. Coquaternion kinematics, which provide hyperbolic rotations, leave the space geometrically unbounded [14]. On the other hand, the structure of biologic organization (module) predisposes the existence of morphologic and functional boundaries.

Only a few data imply the existence of functional links between $PNR$ elements. The character of metabolic chains within biological cells has led some authors to suggest that negative feedback might provide some additional energy to activate biochemical reactions required for a positive feedback function [15]. It could also be confirmed, based on the well-known physiologic interactions between organs, such as a second stage of delivery when active contractions of the uterus increase the amount of the released hormone oxytocin, which by positive feedback increases the strength of the uterus contractions, which in turn further increases the amount of oxytocin and so on; negative feedback would not be able to support the delivery mechanism. Bowel movements and rectal emptying, as well as sexual intercourse, are explicit examples demonstrating that $P^{+}$ provides qualitatively different system actions and requires more energy compared to the $N^{-}$ [16,17]. No data indicating the character of functional links between $R^{\pm }$ and other base elements have been found. Some methods, such as the homotopy perturbation method and Taylor’s series approach, provide highly accurate analytical solutions for nonlinear systems [18,19,20]. Finding relations between base patterns may help to explain mechanisms of differentiation and integration of biologic systems and clarify the role of coquaternions in forming the structure of a functional module.

2. A Mechanical Pendulum as a Model of Functional Relationships Between Base Elements of a Biologic System

A functional structure of $PNR$ base elements is better understood through the phase portraits of ordinary differential equations (ODEs). Linear dynamical models of the base patterns of biologic systems provide a good approximation of feedback circuits. Schematic descriptions of feedback circuits as two-element graphs are still considered the basic tools for understanding the physiology of these regulatory mechanisms.

Feedback responses reflect the ability of a biologic system to return the deviated states back to equilibrium. Equilibrium conditions are known as physiologic constants, such as normal heartbeats, blood pressure, the concentration of electrolytes, hormones, and glucose in the blood, etc. Achieving and maintaining equilibrium, as one of the goals of disturbed systems, usually determines the modeling features to explain the system’s behavior and viability [21,22,23].

Biologic systems are dissipative structures. It means that there is always some amount of energy accompanied by metabolic processes in the form of heat, molecular structures, etc., which cannot be reutilized further by a system. Therefore, the functional dynamics of biologic systems, including the achieving of equilibrium states, presume that the amount of energy needed for synthesizing a biologic matter must include not only the portion, that will be conserved in the created biologic elements but also an amount equal to the dissipated component. In other words, the functional outcome of synthesized biologic cells, tissues, and organs must outweigh the system’s functional decline up to the destruction of morphologic elements as a result of natural physical decay [24,25,26,27]. Physical environment and metabolism itself irreversibly destroy biologic matter; only biologic cells involved in a reproductive cycle provide individual and long-term survival of biologic tissues, organs, and whole organisms. Therefore, the functional stability of biologic systems is not related to achieving and keeping static conditions classically determined as equilibrium states, but a dynamical process prompting permanent input of energy to compensate for natural functional decline and maintain being achieved functionality. The Cell Renewal Cycle (CRC) (not to be confused with the cell mitotic cycle), consisting of the oppositely directed cell destruction and cell proliferation processes, is the simplest physiologic model of metabolism, which generalizes the notion of homeostasis mechanisms as “chasing” keep increasing thresholds levels of equilibrium states [14,28,29,30].

The Gibbs equation $G=H-TS$ is the simplest expression of the fact that, if entropy $S$ of the system is an increasing in time function, energy flow into the system $G$ should also be increasing in time to compensate for the growing entropy S and keep the inner energy of the system $H$ at the functional level. This supports the above statements [31].

There are four types of phase portraits of linear autonomous systems (Jordan normal forms and Poincare diagram) based on the characters of the fixed (equilibrium) points [32,33]. We will consider the “naïve” structural stability of the system based on this classification. Originally $N^{-}$ is represented by a neutrally stable structure [3]. Its phase portrait is a circular trajectory, surrounding the equilibrium point, which is also a phase trajectory. Small disturbances will change the character of trajectories, which may converge toward the equilibrium point or diverge from it. It is natural to consider the disturbing factor as some environmental forces, which will be represented by a diagonal (environment) matrix $E^{\pm }$ added to the matrix of $N^{-}$ (Figure 2). The autonomous systems obtained will show opposite dynamics depending on the signs of the elements of $E^{\pm }$. In this case, classically stable systems shown as converging to the equilibrium phase trajectories eventually will demonstrate no changes in the system’s conditions after achieving the equilibrium states; hence, if staying in these states, the systems will die due to increasing entropy. So, an equilibrium condition can be interpreted as the state eventually leading the system to death. Life is always the fluctuations around equilibrium; this is analogous to a cardiac function measured through the electrical activity of the heart—a straight line (asystole) means cardiac arrest, which is the stop of functioning. On the other hand, unstable systems, which are structurally stable, with diverging dynamics are also not viable because the necessity to increase functional output will eventually meet functional and morphologic incompetence of the system.

$P^{+}$ and $R^{\pm }$ patterns correspond to a saddle and are structurally stable regulatory systems. They contain stable and unstable manifolds [32,34]. Matrices of these operators relative to the same basis determine topologically equivalent systems, which can be obtained by orthogonal transformations or rotations of the plane around the axis through the equilibrium point at 45 deg. Further, it will be shown that the same $S_1$ matrix of reciprocal links $R^{\pm }$, but related to a different operator, is obtained by similarity transformation of $S_2$ matrix of positive feedback $P^{+}$; it changes the basis of the space of biologic variables relative to which matrices are expressed.

If a biologic system has the property to return deviated states to the equilibrium, then some inner forces must exist inside the system, which are assumed to be operated through the activation of the base, $PNR$ elements of the system’s internal structure. To describe this mechanism, consider a simple mechanical pendulum modeling biologic system dynamics. Displacement of the mass from the lowest position, equilibrium, will increase the potential energy of the system, which, after releasing the mass, will cause its movement toward equilibrium, increasing the kinetic and decreasing the potential energy of this system. After reaching the equilibrium, all accumulated by the mass’s displacement potential energy will be transformed into kinetic energy, which reaches its maximum level. If the friction is not considered, the mass will perform small swings around equilibrium with the same amplitude. The swings will cause transformations of potential and kinetic energy to one another until the swings continue [35,36]. A simple pendulum with normalized parameters is described by a classic second-order differential equation.

$$\ddot{x}=-sinx\approx -x+x^3$$

(1)

Equivalent to the system of two ODEs whose variables, $x$—displacement, $\dot{x}=y$ − velocity, which for more accuracy in application to a biologic system is considered with some parameters $b, c>0$

$$\dot{x}= by$$

$$\dot{y}=c(-x+x^3)$$

(2)

The system has three equilibrium points. At zero point its phase portrait is a center, which is a concentric circle. A matrix responsible for this phase portrait is a skew diagonal matrix $S_0=\left(\begin{array}{cc}0& +\\ -& 0\end{array}\right)$. Without friction and external forces, the simple pendulum will perform small swings, and its dynamic expression is analogous to the ($N^{-}$) pattern (Figure 3). The closed trajectories correspond to the energy levels of the system. Total energy of the system $H=-{\displaystyle \frac{x^2}{2}}+{\displaystyle \frac{x^4}{4}}+{\displaystyle \frac{y^2}{2}}$.

The behavior of the pendulum is also well described when friction and external forces are added as factors modifying its behavior [22,30].

For our purposes, we will consider an environmental factor $E^{\pm }$, added to the system (2) in the form of a diagonal matrix with the same signs of coefficients of $E^{\pm }=\left(\begin{array}{cc}a& 0\\ 0& d\end{array}\right), a,d>0; a,d<0$;

$$\dot{x}=ax+by$$

$$\dot{y}=c(-x+x^3)+dy$$

(3)

Parameter $d$ determining friction in classical models ($d<0$) can also provide acceleration ($d>0$), and $a$ of the first equation should have concomitant action on $\dot{x}$ variable because of the same signs of coefficients of $E^{\pm }$.

For simplicity, the system (3) will be considered with the normalized parameters. It also will help to understand and visualize relationships between $PNR$ patterns. The system (3) with normalized parameters has also three equilibrium points: $(x=0, y=0); (x=+\sqrt{2}, y=-\sqrt{2});$ $(x=-\sqrt{2}, y=+\sqrt{2})$ (Figure 4).

Matrix of the normalized and linearized system (3) is

$$\left(\begin{array}{cc}a& b\\ c(3x^2-1)& d\end{array}\right)\approx \left(\begin{array}{cc}1& 1\\ (3x^2-1)& 1\end{array}\right)$$

(4)

At $x=0$ it is presented as a sum of unsteady stellar node matrix $1=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ of environment $E^{+}=\left(\begin{array}{cc}+& 0\\ 0& +\end{array}\right)$ and negative feedback pattern $N^{-}=\left(\begin{array}{cc}0& +\\ -& 0\end{array}\right)$ presented by $S_0=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. The nature of $PNR$ patterns is determined by the signs of entries and determinants of the matrices, not magnitudes of entries, so depending on the context the sign marks as only matrices entries will be used.

The behavior of the system (3) as diverging from equilibrium shows an increase in the total energy supplied by the environment. Physiologic sense of $N^{-}$ and $E^{+}$ base elements acting together depends on the considered time scale. In relatively large periods of time, increasing the total energy of the system can be related to the domination of anabolic processes over catabolic ones when morphologic structures are capable of supporting the system’s function either by proliferation or hyperplasia. For short time intervals, it can be interpreted as rapid movements of the states of the system dictated by the necessity to accelerate metabolism. The behavior, corresponding to the trajectories of the unstable node, is not the only one, which potentially could experience the normal system; a < 0 and d < 0 cause the system to behave as a stable node when the states are converging to the equilibrium. This functional pattern is also physiologically acceptable; for instance, during the physical decline, illnesses, etc., causing exhaustion of the energy of the system, and, if not treated, can lead to death (see Figure 2). For our purposes, we will consider a developing system, which accumulates potential energy in the form of growing and multiplying morphologic elements.

When the energy of the pendulum continues to increase, the character of the movements changes from swings to rotations, which are described by the equations of an upside-down pendulum. It occurs when the increasing energy overcomes some structural threshold of the pendulum. At the points when swings transform into rotations, the system (3) bifurcates (node-saddle bifurcation) and acquires new behavior and a regulatory mechanism. These points are crucial for the system’s behavior. In the phase portrait, they correspond to the homoclinic trajectories, which are two separatrices, forming loops, originating from and coming back to a saddle equilibrium point [35,37] (Figure 5).

As it was mentioned above, (3) has two other, besides zero, equilibrium points. These points at $x=+\sqrt{2}, y=-\sqrt{2}$ and $x=-\sqrt{2}, y=+\sqrt{2}$ correspond to saddles.

The part of the system related to the matrix (4) not affected by the environment $\left(\begin{array}{cc}0& 1\\ (3x^2-1)& 0\end{array}\right)$ has non-degenerate critical point at $x=\pm \sqrt{1/3}$ obtained from $3x^2-1=0$, which are thresholds of potential energy when $N^{-}$ transforms to $P^{+}$. Corresponding phase curves will change the curvatures from convex to concave. From now on the convenient way to describe the system’s development is from the top, non-steady, equilibrium position of the pendulum.

A cubic function $(x-x^3)$ will express the gradient of potential energy $gradV$ of the upside-down pendulum described by the differential equation where the right side of Equation (1) is considered with a positive sign and (2) and (3) will have c < 0. The new equilibrium point for convenience will be at zero coordinates. Corresponding (1′), (2′), and (3′) systems with positive values of the parameters are (Figure 6).

$$\ddot{x}=+sinx\approx +x-x^3$$

(1′)

$$\dot{x}= +by$$

$$\dot{y}=c(x-x^3)$$

(2′)

$$\dot{x}=ax+by$$

$$\dot{y}=c(x-x^3)+dy$$

(3′)

Linearized system (2′) at $x=0, y=0$ is a saddle. The cubic expression for the gradient of the potential energy of the system ${\displaystyle \frac{dV}{dx}}=x-x^3$ will determine closed-phase trajectories because of the prevalence of $-x^3$ when $x$ (potential energy of the system) continues to increase. At $x=\pm \sqrt{1/3}$ obtained from $\left(x-x^3\right)=0$ curvatures of the closed curve will change according to the hyperbolic and elliptic structures of the curve, which will replace each other (see Figure 5).

The behavior of the system without the interference of the environmental factors (2′) shows that after moving to a higher energy level, the system’s conditions may oscillate, confining two-phase portraits of $N^{-}$. Metabolic pathways, shown as negative feedback trajectories, surrounding two other negative feedback curves, require more energy than each of the two banded-together subsystems. The surrounding two systems $N^{-}$ loop is separated from these systems by homoclinic trajectories, which are separatrices, originating from the hyperbolic equilibrium point (see Figure 5). Related transformation mechanisms $N^{-}\to P^{+}\to N^{-}$ reflect a special property of a biologic system to activate its inner energy sources and switch from the lower energy $N^{-}$ pattern to the higher energy-consuming $P^{+}$ function and further on to $N^{-}$. This property, expressed through the cubic function ${\displaystyle \frac{dV}{dx}}=\pm (x-x^3)$, is termed in this work a biogenic active property (BAP) of a biologic system.

Environment $E^{+}$ substantially changes the character of phase trajectories of the system. When $\left|x\right|>\sqrt{1/3}$, the normalized system (3′) becomes $\left(\begin{array}{cc}+& 0\\ 0& +\end{array}\right)+\left(\begin{array}{cc}0& +\\ -& 0\end{array}\right)$, and it is easy to see that the unstable structure of the environment $E^{+}$ transforms closed phase trajectories of negative feedback $N^{-}$ surrounding the other two $N^{-}$ curves into the unstable node. Action of environment $E^{+}$ added to the system (2′) will make it diverge as in the case when the system is regulated by $N^{-}$ and $E^{+}$ patterns before it bifurcates. Phase portraits (2′) and (3′) also show that closed trajectories originated from $P^{+}$ and confining two $N^{-}$ subsystems become divergent. If this regulatory mechanism continues developing without structural changes, it eventually will destroy the system (Figure 7).

It seems that nature has found a means to solve this problem: first, by splitting the $P^{+}$ operator and the space of variables into two parts, and then integrating the obtained components. The eigenvectors of the two one-dimensional subspaces, after splitting, have become expressed relative to a new basis, which are combinations of the previous basis elements. The way the separation of the elements is obtained makes the analytical description, as well as the physiologic process of differentiation not trivial. Moreover, it would also require reorganizing morphologic and functional elements of the system. The process of morphologic adaptation, in fact, begins simultaneously with functional changes in the system, so by the time when qualitative changes have occurred, morphologic elements will be ready to adapt their structure to the upcoming events.

$P^{+}$ transforms to an equivalent $R^{\pm }$ pattern by conjugation diffeomorphism h: $S_{1 }=h{S}_2{h}^{-1}$; $h\in GL(2,\mathbb{R})$. The importance of this transformation for the system’s stability lies in the property of the obtained matrix $S_1$ to have a diagonal view with real elements (eigenvalues), thus to be presented as a direct sum of two operators acting on one-dimensional subspaces. Moreover, the similarity transformation changes the initial basis $\left\{e_1,e_2\right\}$ to $\{f_1, f_2\}$ relative to which the $S_1$ matrix of $R^{\pm }$ is expressed. The new basis $\{f_1, f_2\}$ is a linear combination of $e_i$ basis elements: $f_1=e_1+e_2$, $f_2=e_1-e_2$. Combinations of the basis elements and opposite signs of eigenvalues make representing subsystems act as separate units with opposed functionality. Matrix $S_1$, representing the reciprocal links $R^{\pm }$ pattern, is a direct sum of two operators $S_{1 }={\phi }_1\oplus$ ${\phi }_2$ in the space of biologic variables $V$, which is a direct sum of two one-dimensional subspaces $V_1$ and $V_2$ spanned by the new basis elements ${<f}_1>\oplus <f_2>$. As a result, the system can hold total energy, which becomes distributed between two relatively simpler independent subsystems. Important property of $R^{\pm }$ is providing the “special” functional links between vectors $S_1({v_1, v_2)\in V}_{1}x$ $V_2$ satisfying $v_1{v}_2=c$, where $c$ is a constant. Corresponding hyperbolas are trajectories of the same energy levels.

What the pendulum model implies is that accumulated energy “helps to choose” the right regulatory mechanism (functional pattern) to manage the current system’s condition. Thus, the properties of four base elements organized in a sequence will determine the global functional structure and dynamics of the system, depending on the available energy. That functional patterns can be organized in a sequence, in fact, reflects the global tendency of the system to use increasing amounts of energy to maintain continuity of metabolism and stability of the performing functions. $N^{-}$ and $P^{+}$ related mechanisms in the pendulum model were modified by environmental factors $E^{\pm }$.

3. The Structure of the Base Functional Patterns Determines Nonequilibrium Dynamics of Biologic Systems

Base functional elements of the system related to the environmental pattern present basis elements of coquaternions (the coquaternions’ ring), whose elements span pseudo-Euclidean 4-space over $\mathbb{R}$. The heterogeneity of the space ${\mathbb{R}}_2^4$, determined by the structural differences in the base elements, in turn, determines the character of relationships between the variables $x$ and $y$. The $N^{-}$ and $P^{+}$ patterns determining changes in the system’s conditions cannot split $x$ and $y$ variables. In other words, the well-known negative and positive feedback circuits can’t manage involved in these mechanisms subsystems separately. If the current condition of one subsystem changes, the condition of another subsystem will also change, obeying the feedback action of the first subsystems. Each condition of the system is a pair $(x,y)$ considered as a whole character. Only virtually are they separate, independently acting characters. For instance, pituitary and thyroid glands are anatomically separate organs with their own functions. Roughly pituitary produces thyroid stimulating hormone TSH and thyroid gland-T3 and T4 hormones. Any changes in the concentration of thyroid hormones will evoke a negative feedback response in the pituitary and change in concentration of TSH. In turn, it will stimulate or inhibit the actions of the thyroid. These organs act as the whole functional unit. None of the known physiologic mechanisms makes them act independently. The same is true regarding the strength of the uterus contractions and the posterior pituitary responsible for the release of oxytocin during labor. Only together are these organs capable of providing the uterus with the contractile forces required for delivery, thus, to perform the system’s action generated and regulated by a positive feedback mechanism.

This simple from the first glance observation lies on the property of $S_0$ and $S_2$ ($N^{-}$ and $P^{+}$ patterns, respectively) to be the matrices of complex and split-complex structures on the spaces of variables $\left(x,y\right)\in$ $\mathbb{R}$ $\times$ $\mathbb{R}$ and operators $S_0(x,y)$ and $S_2(x,y)$ determining phase flows as one parameter $\left(t\right)$ groups of diffeomorphisms $S_0^t:\mathbb{R}\times V\to V$; $S_2^t:\mathbb{R}\times W\to W$. Decomplexification allows these structures to be presented as direct sums of subspaces and operators. This construction, in fact, doubles the dimensionality of the real spaces where physiologic variables and operators are being measured.

$V^{\mathit{R}\left(\mathit{i}\right)}=V_1\oplus {iV}_2$ and $W^{\mathit{R}\left(\mathit{j}\right)}=W_1\oplus {jW}_2$ are direct sums of the spaces of splitors [8], where $V_2(x,y)$ and $W_2(x,y)$ are the spaces for $N^{-}$ and $P^{+}$ patterns and environmental spaces $V_1(x,y)$ and $W_1(x,y)$ for $E^{\pm }$ pattern. Now the spaces of physiologic variables with the environment can be considered as the whole units. The existence of the environmental component $E^{\pm }$ becomes a natural complement to $N^{-}$ and $P^{+}$ resulted in a doubling of the dimensionalities of the spaces. It was an environmental element $E^{\pm }$ formally added to occupy the real parts of the obtained subspace.

$A^{\mathit{R}\left(\mathit{i}\right)}=B\oplus iC$, $D^{\mathit{R}\left(\mathit{j}\right)}=F\oplus jG$ are complex and split-complex matrices with respect to $\left\{e_1,e_2,i{e}_1,{ie}_2\right\}$ and $\left\{e_1,e_2,j{e}_1,{je}_2\right\}$ bases of operators $f:V\to V$ and $g=W\to W$, respectively, where $B=aE$, $C=b{S}_0$ and $F=cE$, $G=d{S}_2$; $a,b,c,d\in \mathbb{R}$. $E$, $S_0$, $S_2$ are matrices corresponding to the environment and negative feedback and positive feedback, respectively. These structures can be written in the following forms: $a\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\oplus b\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$ for complex matrices; $c\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\oplus d\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ for split-complex matrices.

As it was mentioned above, $S_0$ and $S_2$ are irreducible second-order matrices relative to the standard basis.

Split-complex matrix $S_2=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ has its conjugate $S_1=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$, and this transformation makes regulatory structure reducible to one-dimensional operators ${\phi }_1$ and ${\phi }_2$, whose eigenvalues are of opposite sign. These operators form a direct sum $S_1={\phi }_1\oplus {\phi }_2$, which makes them act separately on one-dimensional invariant subspaces ${<\phi }_1{\acute{x}}_1>={\acute{W}}_1$, ${<\phi }_2{\acute{x}}_2>={\acute{W}}_2$; $\acute{W}={\acute{W}}_1$ ⊕${\acute{W}}_2$ and also on pairs of functionally related variables satisfying ${\acute{x}}_1{\acute{x}}_2=const.$ (see Figure 1). These spaces are over $\mathbb{R}$. ${<\phi }_1\mathbb{R}>=\mathbb{R}$, ${<\phi }_2\mathbb{R}>=\mathbb{R}$, $\acute{W}\approx \mathbb{R}$ ⊕$\mathbb{R}$. The new basis elements of ${\acute{W}}_1$ and ${\acute{W}}_2$ are $f_1=e_1+e_2$, $f_2=e_1-e_2$, respectively.

The matrix of a complex structure $S_0=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$ also has its conjugate as a diagonal matrix with complex eigenvalues ${\tilde{S}}_0=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right)$. It is a direct sum of two complex operators ${\tilde{S}}_0={\omega }_1\oplus {\omega }_2$, acting on complex eigenvectors of one-dimensional invariant subspaces associated with the new complex basis vectors along coordinate axes ${\acute{y}}_1$ and ${\acute{y}}_2$: ${<\omega }_1{\acute{y}}_1>={\acute{V}}_1$, ${<\omega }_2{\acute{y}}_2>={\acute{V}}_2$; $\acute{V}={\acute{V}}_1$⊕ ${\acute{V}}_2$. These spaces are over $\mathbb{C}$. ${<\omega }_1\mathbb{C}>=\mathbb{C}$, ${<\omega }_2\mathbb{C}>=\mathbb{C}$, $\acute{V}\approx \mathbb{C}$ ⊕ $\mathbb{C}$.

New basis for ${\acute{V}}_1$ and ${\acute{V}}_2$ is $g_1=(e_1+{ie}_1)$, $g_2=(e_2-{ie}_2)$. These are not the only combinations acceptable for the new basis elements from ${\mathbb{C}}^2$. For instance, other combinations can be $g_1=(e_1+{ie}_2)$, $g_2=\left(e_2-{ie}_1\right).$ Important, that real parts are from ${\mathbb{C}}^2\cap {\mathbb{R}}^2$.

The spaces ${\acute{W}}_1,$ ${\acute{W}}_2,$ ${\acute{V}}_1,{\acute{V}}_2$ are disjunctive, satisfying the condition ${\acute{W}}_1\bigcap$ ${\acute{W}}_2\bigcap$ ${\acute{V}}_1\bigcap {\acute{V}}_2=0$, so that a direct sum ${\acute{W}}_1\oplus$ ${\acute{W}}_2\oplus$ ${\acute{V}}_1\oplus {\acute{V}}_2$ is a four-dimensional space and linear combinations of the vectors from this space allow considering the sums ${\acute{W}}_1+{\acute{V}}_1$ and ${\acute{W}}_2+{\acute{V}}_2$, which for ${\tilde{S}}_0$ and $S_1$ operators in a matrix form reads

$$S=\left(\begin{array}{cc}\left(\begin{array}{cc}1& -1\\ 1& 1\end{array}\right)& 0\\ 0& \left(\begin{array}{cc}-1& 1\\ -1& -1\end{array}\right)\end{array}\right)$$

It is obvious that the initial space has become split into the two functionally opposite subspaces regulated by ${(E}^{+}-N^{+})$ and ${(E}^{-}+N^{-}$)

For the real form of ${\tilde{S}}_0$, ${\tilde{S}}_0^R=\left(\begin{array}{cc}\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)& 0\\ 0& \left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\end{array}\right)$, there is a conjugation matrix C∈${SO}_2$, which transforms it to a skew diagonal form ${\check{S}}_0=C^{-1}{\tilde{S}}_0^{R}C$ =$\left(\begin{array}{cc}0& \left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)\\ \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)& 0\end{array}\right)=\left(\begin{array}{cc}0& -E\\ E& 0\end{array}\right)$.

Now it is easy to see that the obtained negative feedback pattern ${\check{S}}_0$ is presented by compositions of elements, which are integrated from the simpler patterns. This construction defines a hierarchy between blocks considered as the single not structured regulatory elements operating on a pair of variables and elements of the blocks providing differentiated actions on single variables. Thus, the matrix blocks transform pairs of variables as a whole element; after that, each variable is managed separately as a discrete element, depending on the inner structure of the considered block.

${\check{S}}_0$ works as $S_0$ pattern of negative feedback $N^{-}$ whose elements are the skew diagonal unit matrices organized in 2 × 2 blocks with opposite entries. ${\check{S}}_0$ will make the related 4-vector involved in the next $N^{\pm }\to P^{+}\to R^{\pm }\to N^{\pm }$ cycle. The structure of the two inner functional cycles $N^{-}$ and $N^{+}$ related to the split elements will be finalized in the formation of two orthogonal coquaternions.

This construction also determines the evolution of the systems in two different directions: by forming the structural complexes from the existing elements, and, at the same time, splitting the element into congruent simpler components.

Actions of environment $E^{\pm }$ (system 3′) will provide the evolution of the system’s states as diverging phase trajectories result in the following sequence of regulatory mechanisms: center, focus (unsteady node), and then, saddle with homoclinic trajectories included.

The images of two unsteady nodes bounded by closed curves (or curves of unsteady nodes in case of environmental actions) correspond to the subsystems that would be stemmed and developed after the bifurcation. The neighborhood of a saddle point presupposes the formation of these subsystems. The four separatrices virtually split the system and demonstrate equal contributions of two subsystems, elliptic and hyperbolic, in the evolution of the whole system. The right upper separatrix shows a simultaneous increase in potential and kinetic energies ($x$ and $\dot{x}$ variables), moving the system’s conditions to a higher functional level. The phase portrait of the system after the splitting also corresponds to the saddle equivalent to the image of $P^{+}$ rotated on 45 degrees counterclockwise (see Figure 1). Two one-dimensional subsystems obtained after the transformation will be defined by new variables, which are obtained as linear combinations of the previous ones, $(x,\dot{x})$. $f_1=e_1+e_2$, $f_2=e_1-e_2$ as the new basis vectors are orthogonal to each other as eigenvectors of one-dimensional subspaces. Considered as new variables ${\breve{x}}_1$ and ${\breve{x}}_2$ with their velocities added to the system as variables $\dot{{\breve{x}}_1}$ and $\dot{{\breve{x}}_2}$, the formed subsystems will be relatively independent and regulated by orthogonal operators ${\phi }_1$ and ${\phi }_2$. Obtained $R^{+}=\left({\breve{x}}_1,\dot{{\breve{x}}_1}\right)$ and $R^{-}=\left({\breve{x}}_2,\dot{{\breve{x}}_2}\right)$ systems are similar to the initial system $(x, \dot{x})$ as potential and kinetic energy variables. Two connected by the linking diffeomorphisms subsystems $R^{+}$ and $R^{-}$ also, present a system $R^{\pm }$. As a whole functional structure, it will follow the same nonequilibrium changes as the initial system. Thus, the closed regulatory cycle $(N^{-}\to P^{+}\to R^{\pm }\to N^{\pm }\to )$ = $(N^{-}\to N^{\pm })$ is obtained when after the splitting two subsystems hold the wholeness of the system’s structure due to integrating the subsystem’s reciprocal links (diffeomorphisms). The first $N^{-}$ originates from the last $R^{\pm }$ pattern of the previous cycle, so it inherits the basis relative to which the corresponding $S_0$ and following matrices are expressed until $P^{+}$ splits again. From the split points, there are two directions of how the progeny system will develop—two split subsystems can act as independent units (systems) and as an integrated system. As a system, it can develop independently, following previously described mechanisms of transformations. Physiologically the first ones provide a biologic system with the mechanism of differentiation, while the latter with the integration.

4. Coquaternion Module Dynamics in Transformations of the Base Patterns

All coquaternion basis elements are involved in a $N^{-}\to P^{+}\to R^{\pm }\to N^{-}$ chain as regulatory patterns. The scalar element $E^{\pm }$ added to each part provides a smooth transformation or transition of the patterns, depending on thresholds and the specific character of the development. These patterns may form combinations with each other. For example, a combination of pure $P^{+}$ and $R^{\pm }$, is able to deform $P^{+}$-related diffeomorphisms, not changing its hyperbolic nature. Physiologically, most probably it would be preceded by $N^{-}$ and involved in regulations, especially in developing organisms after the system bifurcates. $R^{\pm }$ is a diffeomorphism linking neighbors, being split, functional systems in a manner preserving the dot product of these systems, $xy=c$. It determines horizontal relations between the two newly born developing characters. On the other hand, the two subsystems determining these characters pursue their own coquaternion cycles, so they develop vertically, inside the system. Thus, evolved from the initial system, a functional module determines a two-level hierarchy, or two orthogonal hypersurfaces presented by coquaternions. Formally speaking, the obtained two pathways organized as orthogonal circles form a torus (algebraic torus under some values of parameters) where each circle trajectory is a sequence of coquaternion elements.

The geometry of coquaternions is related to ${\mathbb{R}}_2^4$ space and its indefinite metric structure is determined by the algebra of the basis $\{1,i,j,k\}$ elements. When the basis is represented by the above-discussed matrices $\{E,S_0,S_1,S_2\}$, the matrix determinants, $\det \left(S_i\right)$, will determine diagonal elements of the Gram matrix of quadratic form and (2,2) metric index of the 4-space of coquaternions [7,8,38]. For the given coquaternion $q=wE+x{S}_0+y{S}_1+z{S}_2$ the scalar product on one-forms is a quadratic form $<q,q>=w^2+x^2-y^2-z^2$. The value of the quadratic form $<q,q> \in \mathbb{R}$ is determined by one-forms and they will not change the metric index. Changing values of one-forms will only “deform” the geometry of the surfaces leaving the metric index the same. Thus, determinants of the matrices of the basis elements will determine the main structural characteristics of the functional module of the normal biologic system.

No other than cℍ structure with the basis obtained from the three forms of feedback exists, if maintaining the wholeness of biologic systems’ structure is determined, first, by splitting and, second, by integration of split components when the system’s conditions reach some functional thresholds during development. Maintaining a normal system structure is functionally equal to the system’s evolution in its narrow sense, considered as a process escaping equilibrium states and providing functional stability and structural steadiness through the transformations of the base regulatory patterns. Closed cℍ structure, as an algebraic noncommutative ring, confirms or suggests the closed structure of relationships among the base elements. Split-octonions, as the next algebraic construction, lose the distributive property of their elements and become unstable as a functional unit in terms of the reproductivity of their regulatory structure in biologic systems. So, four-dimensional spaces provide structural limits to present a functional module as an invariant regulatory structure of biologic systems.

$S_2\left(j\right)$ and $S_1\left(k\right)$ (in parentheses are formal cℍ basis elements) will determine orthogonal coordinates and geometric images of hyperbolic hyperboloid (hyperboloid of one sheet). It is expected that the system’s dynamics will demonstrate similar images in ($S_0\left(i\right)$, $S_2\left(j\right)$) and $\left(S_0\right(i)$, $S_1\left(k\right))$ planes because of the automorphism of $S_2\left(j\right)$) and $S_1\left(k\right)$ operators, determining closed trajectories obtained from the sections of hyperbolic hyperboloid perpendicular to $S_0\left(i\right)$ axis. Sections through $x=\pm \sqrt{1/3}$ will determine non-degenerate critical points as points on the circle where the hyperbolic hyperboloid transforms into the sphere. Closed trajectories related to the pendulum rotations will be projections of the three-dimensional image onto the plane of $(x,y)$ variables. Each closed surface corresponds to a certain energy level as the value of the associated quadratic form, whose metric will undergo transformation from $(++--)$ to $(++++)$. This can be seen from the linearized matrix $\left(\begin{array}{cc}0& 1\\ (1-3x^2)& 0\end{array}\right)$, having alternating determinant values around critical points, therefore, making changes in the metric index (Figure 8).

Thus, the cubic function makes unbounded coquaternion surfaces bounded and closed by portions of the spheres. It demonstrates additional features of biogenic active properties of biologic systems. The system becomes capable of confining the energy at a certain conditionally stable level and acts as an autonomous system also maintaining the activities of two other systems. It means that the coquaternion module contains integrative properties in its structure, allowing the obtained after splitting subsystems, to be regulated as a whole unit by $N^{-}$ (closed trajectory).

On the other hand, the action of the environment can make the surface of the hyperbolic hyperboloid diverge markedly from the zero point, which must suggest the necessity of earlier system transformations, including splitting (Figure 9). Metric transformations can be a normal part of the system’s behavior, allowing the system to maintain its functionality adequate to the available energy, preventing the system’s destruction. When nonequilibrium states are considered normal system dynamics, actions of the environment will keep the system moving to the higher functional levels through the thresholds when regulatory mechanisms switch to apoptosis, cell proliferation, and eventually to the reproduction of the whole organisms.

5. Results

It is implied that, in order to maintain the continuity of metabolism, the sequence of transformations of the base elements as regulating metabolism patterns must be finalized in a “special” regulatory pattern, which will create a new germ: two separate characters as autonomous systems, each potent for the development and at the same time congruent to be integrated into another system. This pattern is presented by a reciprocal operator, which is a direct sum of two operators acting on one-dimensional subspaces. The similarity transformation by which this pattern is obtained splits the regulatory structure of the new system into two relatively independent mechanisms, dividing the system into two subsystems. This provides a biologic system with continuity of functional regulatory mechanisms by creating separable inner morphologic elements presented by two subsystems.

Continuity of biochemical reactions through transformations of PNR base physiologic patterns is a substantial feature maintaining the viability of biologic systems, thus, determining homeostasis mechanisms. Considering metabolism as a process of obtaining energy for chemical transformations and changes in the system’s conditions, homeostasis will make metabolic flows continuous and “avoiding” equilibrium states. On a cellular and organ level, homeostasis is provided by the continuity of symmetric and asymmetric cell divisions [14,39,40] involved in CRC.

Because of splitting the initial system into two subsystems and identifying split elements (subsystems) with the progenitor system, the way the uniform structure, a biologic system, is obtained also determines a functional hierarchy between the systems. There are only four pieces of a one-dimensional manifold (saddle) associated with the coordinate axes where the split systems are totally independent. The areas between them are filled with the hyperbolic curves of “reciprocal” diffeomorphisms between subsystems satisfying $xy=const$.

Each hyperbolic trajectory lies in a certain energy level of the system. The character of the trajectories also shows mutual transformations of the kinetic and potential energy of the components. Reciprocal links, in fact, are a mechanism of integration of biologic systems that originated from common roots before splitting. A demonstrative example is the clot-formation and clot-degradation subsystems regulating the viscosity of the blood. The function of each subsystem is provided by cascades of biochemical reactions acting in a reciprocal manner: when the clot-formation subsystem is active, making the viscosity of the blood high enough to block the arteries by forming clots, the clot-degradation subsystem will act inversely to compensate for the functions of the overactive clot-formation system and prevent harmful consequences of not keeping the rheology of the blood within normal physiologic ranges.

Coquaternion structure is a logical sequence of the four base elements organized in a functional cycle, forming the vertical hierarchy of developing systems. The closed algebraic structure of coquaternions implies the wholeness of functional relationships between base regulatory elements, forming the structure of a functional module of biologic systems.

The obtained results can be summarized as follows: the strategy of biologic systemogenesis is aimed at the formation of discrete units with highly integrative properties.

6. Discussion

A biologic system is a dissipative structure and an equilibrium state, when conditions are static, not changing over time, and might not characterize the system’s stability and viability. A viable system dynamic is always fluctuations (!) of the conditions around the equilibrium. Only on a large scale of measured quantities fluctuations are not detectable, therefore, conditions are considered stable, not changing in time characters. Moreover, biologic systems have some stages of maturation that should displace the normal ranges of physiologic constants, and the states initially considered normal when the system is immature are becoming abnormal and out of normal ranges with aging. The cell renewal cycle (CRC), which, in fact, should be considered as the System Renewal Cycle (SRC), validates this statement by demonstrating continuous renewal processes of biological cells, tissues, and organs, renovating “old” and malfunctioning systems.

The Gibbs equation formally implies the dissipative property of biologic objects by including entropy in the equation, which would increase the total amount of energy required for the system to maintain its structure and function. Permanently growing entropy increases the system’s malfunction which, in turn, will require an increasing amount of energy input to maintain the functionality.

In order to describe the “nonequilibrium” behavior of biologic systems, the simple physical analog, a mechanical pendulum, has been considered as a model.

The function of the mechanical pendulum is determined by the potential energy of the mass displaced from the equilibrium. Until additional energy is added to the pendulum, it will perform simple swings around the low-equilibrium position. When the pendulum attains an amount of energy overweighting its structural threshold, the swings change to rotations. Behavioral patterns described by ODE will demonstrate two qualitatively different regulatory mechanisms determined by mutual transformations of potential and kinetic energy and the total amount remaining conserved. Corresponding to the pendulum dynamics phase trajectories are center and saddle. These events occur according to the Hamiltonian mechanics, which, in small, mimics biologic processes.

The well-known physiologic property of biologic systems to move displaced conditions to the normal, equilibrium states was the determining factor in formulating the modeling conditions. Negative $N^{-}$ and positive $P^{+}$ feedback circuits observed in biologic systems have been found to be similar to the pendulum mechanisms determining behavior. If additional energy is added to the pendulum, it causes transformations of phase trajectories from the center to the saddle and changes the character of the pendulum movements from swings to rotations.

Unlike mechanical systems, a biologic system is able to activate its inner sources and switch regulatory mechanisms from $N^{-}$ to $P^{+}$. This feature was named in this work as a biogenic active property (BAP) of a biologic system and the cubic function has provided this modeling feature. An environmental operator in the form of a diagonal (identity) matrix was added to the system of ODE. Energy levels become not quantized as in the case of “conditionally” steady $S_0$ operator of negative feedback. Environmental operators make transformations of the system’s conditions to be smooth diverging from equilibrium phase trajectories, moving the system’s conditions toward a functional threshold.

A nonlinear element in the form of a cubic function is key in determining the first node–saddle bifurcation and further dynamics related to the formation of closed energy trajectory confining two $N^{-}$ curves. The dominance of the cubic summand during increase in potential energy changes regulatory patterns from $P^{+}$ to $N^{-}$, which returns back when potential energy decreases, and the kinetic energy component becomes the leading cause in changing the character of movements and keeping the trajectory closed. Thus, the curve, being initially hyperbolic, becomes elliptic, and then changes its shape again when approaching saddle equilibrium. The changes continue in the opposite direction until the curve becomes closed. A closed trajectory corresponding to the constant energy level becomes diverging from the saddle equilibrium in the form of an unstable node when an identity matrix is added, making the environment a factor providing growth of the total energy of the system.

Similarity transformation $S_2\to S_1$ preserves the matrix determinant that suggests the operation as requiring minimum energy. Hence, by the time node–saddle bifurcation occurs, the obtained $P^{+}$ regulatory pattern would simultaneously transform to $R^{\pm }$ and both under the actions of the environment would evolve further, following diverging trajectories. Similarity transformation resulted in a conjugation diffeomorphism that splits the operator of positive feedback into a direct sum of two operators of reciprocal links. The physiologic importance of this transformation lies in the distribution of the conserved total energy between two subsystems and the property of obtained regulatory structure to manage each one-dimensional subsystem independently. On the other hand, the structure of $R^{\pm }$ allows the newly obtained variables to be regulated, following reciprocal feedback between them such that to keep the integrated energy level on the constant level. Conjugating diffeomorphism provides further development in two simultaneously evolving directions—vertically by pursuing differentiation of obtained morphologic elements and horizontally integrating differentiated elements.

Closed planar trajectories, corresponding to the pendulum model, will be transformed into the closed surface(s) in the case that the matrices of the environment $E^{\pm }$, negative feedback $S_0$, positive feedback $S_2$, and reciprocal links $S_1$ will be considered as forming a basis for coquaternion, representing a functional module of biologic systems. Functional structure, corresponding to the closed surfaces, is a fragment of dynamical changes along geodesics, demonstrating equipotential conditions related to a certain constant level of the total energy of the system. Physiologically, these conditions are “equilibrium” points of a constant value of the difference between anabolism and catabolism measured for conservative systems at a certain time. The equivalence of $S_2$ and $S_1$ operators provides symmetry of coordinate axes associated with quadratic form $–\left(y^2+z^2\right)$ and the cylindrical surface of the hyperbolic hyperboloid. Its transverse sections are circles (ellipsoids) determined by conjugating $S_2$ and $S_1$ variables diffeomorphisms. A not-bounded surface of a hyperbolic hyperboloid theoretically is a point when catabolism dominates anabolism, and quantitative difference remains at a certain level, when both components increase or decrease simultaneously. So, these surfaces are equinormal points when opposite metabolic components increase or decrease simultaneously. When the sum of the absolute values of the total amount of potential and kinetic energy remains at the same level, the corresponding surface becomes equipotential and closed. Equipotential conditions must also cause fluctuations of potential and kinetic energy components, whose sum will be kept at a constant level, according to the parameters of the surface. To maintain the same total energy level (=closed surface), hyperbolic hyperboloid determining coquaternion metric index $(++--)$ undergoes transformation to $(++++)$, corresponding to the sphere and backward until closed surface is obtained. The environment will make the closed surface sliced off though. Slices will diverge from the zero point. Although the environment makes system dynamics and related trajectories divergent and unstable, morphologic structure will determine functional limits, serving as thresholds when regulatory mechanisms switch the transforming system to the next stage of development unless it splits, and a new cycle begins. From this, it directly follows that biologic systems evolve through the formation of discrete, integrable units.

7. Conclusions

Understanding the nature of biologic systems through the geometry of cℍ is a new methodology for investigating biologic objects. Biologic systems are considered complex structures where even between anatomically distinguishable elements, such as cells, tissues, and organs, functional links remain not strictly determined. Medical and biologic processes will become more observable by introducing cℍ as a space of algebraic elements, providing biologic systems with the structure of a functional module. Still being a rough approximation of real biologic mechanisms, the algebraic structure of cℍ characterized by an indefinite metric structure that may provide new physiologic criteria for practical investigations. The (2,2) metric index seems to be related to the opposite physiologic directions provided by feedback regulatory patterns—catabolic and anabolic. In the formulation of the physiologic conditions for mathematical modeling, equilibrium states should be determined not as a final and desirable outcome for the system’s behavior, but functional decline, disease, or death. Continuity of metabolism presupposes the reproducibility of biologic elements only after they become functionally incompetent and reach stable equilibrium conditions. Increasing energy demands by regulatory mechanisms can also be the subject of practical investigations, determining the functional and morphologic potency of the systems. Further theoretical and practical investigations should also be directed toward understanding how physiologic combinations of base regulatory $PNR$ elements contribute to the functional features and stability of BS that may further elucidate their role in homeostasis mechanisms and in a hierarchical structure of the carrier space of splitors.