Formulating a Mathematical Model for Living Systems

1. Introduction

In 1978, Prigogine delineated dissipative structures [1], which exhibit several traits shared with living systems, such as self-assembly [2,3,4]. However, these parallels do not encompass all the characteristics of life. Many dissipative systems exist that are not living organisms, such as cyclones and turbulent flows. Therefore, it is essential to explore additional tools for modeling life, focusing on characteristics unique to living organisms rather than generic dissipative structures.

To distinguish living organisms from other dissipative systems, we must identify a defining property exclusive to life. I propose that a key characteristic of living organisms (referred to as living systems) is their ability to observe other systems in their surroundings. This observation process is intricately linked to the concept of “measurement” in quantum mechanics.

The acquisition of information about the physical universe heavily relies on the process of “measurement” [5]. Howard Pattee has emphasized that “measurement” processes must inherently connect with living systems [6].

2. Mathematical Model

2.1. Quantum Dynamical Dissipative Systems

Most mathematical concepts are founded on set theory. To mitigate the risk of paradoxes in the model, it is crucial to adhere to the notion of a ’pure set,’ excluding entities like proper classes. Therefore, I assume $\mathcal{M}$ to be a Grothendieck universe, defined as the universe comprising only ’pure sets’ [7]. In this context, $\mathcal{H}$ represents the set of all linear maps from $\mathcal{M}$ to $\mathbb{R}$, and $\mathcal{I}\subseteq \mathbb{R}$ is a finite set consisting of specific (finite) values $h\left(x\right)$ for some $x\in \mathcal{M}$ and certain linear maps $h\in \mathcal{H}$. This assumption of a finite set is based on the hypothesis that living systems, including humanity, cannot yield infinite values when measuring anything.

Let ${\mathcal{M}}_H\subseteq \mathcal{M}$ be a Hilbert space equipped with a norm $∥x∥=\sqrt{\langle x,x\rangle }$ and a distance function $d(x,y)=∥x-y∥$ defined for each $x,y\in {\mathcal{M}}_H$. Then $M_H$ is also a Banach space, and the set of all (Lipschitz) continuous linear maps from ${\mathcal{M}}_H$ to $\mathbb{R}$ is the dual space of ${\mathcal{M}}_H$ (denoted by ${\mathcal{M}}_H^{*}$). ${\mathcal{M}}_H^{*}$ is also a Hilbert space [8,9].

Let ${\mathcal{I}}_H\subseteq \mathcal{I}$ be a set of finite eigenvalues of operators on ${\mathcal{M}}_H$, then ${\mathcal{I}}_H$ is finite. Let $X\subseteq {\mathcal{M}}_H$ be a closed subset of ${\mathcal{M}}_H$, $S_t$ be an evolution operator, and $B\subseteq X$ be an absorbing set. Then $(X,S_t,B)$ is a dynamical dissipative system [10]. According to Riesz-Fréchet Representation Theorem [8], for each $f\in X^{*}$, there is a unique $x\in X$ such that $f\left(y\right)=\langle x,y\rangle$ for all $y\in X$. We denote f as $\langle x|$, y as $|y\rangle$, and $\langle x,y\rangle$ as $\langle x|y\rangle$. Additionally, we define a set $U_X$ of evolution operators represented by

$$\begin{array}{cc}\hfill U_{t-t_0}:X& \to X,\hfill \\ \hfill |x_{t_0}\rangle & \mapsto |x_t\rangle ,\hfill \end{array}$$

where $t,t_0\in {\mathbb{T}}_X\subseteq {\mathcal{I}}_X$ and $t\ge t_0$. The operators in $U_X$ satisfy the following properties: $U_{t-t}|x\rangle =U_0|x\rangle =|x\rangle ,U_{t-t_1}\circ U_{t_1-t_0}|x\rangle =U_{t-t_0}|x\rangle$, where ∘ is the composition of operators. These operators are known as the time-evolutions or the propagators [9]. Each element $U_{t,t_0}$ of $U_X$ can be represented by the pair $(t,t_0)$.

An absorbing set of X is a closed set $B_X\subseteq X$ which has the following property: for any bounded set $D\subseteq X$, there exists $t_D\in {\mathbb{T}}_X$ such that $U_{t-t_0}\left(D\right)\subseteq B_X$ for all $t,t_0\in {\mathcal{I}}_X$ with $t\ge t_D\ge t_0$, where $U_{t-t_0}\left(D\right)=\{\left.U_{t-t_0}|x\rangle \right|x\in D\}$.

A system $(X,U_X)$ is called a quantum dynamical system, and $(X,U_X,B_X)$ is called a quantum dynamical dissipative system, where X is a closed subset of a Hilbert space, $U_{t-t_0}$ is the time evolution operator, and $B_X$ is an absorbing set of X.

2.2. Living Systems

A living system has properties of a dissipative structure, therefore it is a quantum dynamical dissipative system. Let $(X,U_X,B_X)$ be a quantum dynamical dissipative system, and let $X_Y$ be a subset of X. Suppose that every element of $X_Y$ can be represented as $x_{ij}$, where $i,j\in {\mathbb{Z}}_{+}$ and $i\le j$. Let * is a binary operator defined on $X_Y$ such that $x_{ij}*x_{jk}=x_{ik}$ whenever $x_{ij},x_{jk},x_{ik}$ are in $X_Y$. The set $X_Y$ is not necessarily closed under *, as there may exist $x_{ab},x_{bc}$ in $X_Y$ such that $x_{ac}$ does not belong to $X_Y$.

The distinctive characteristic of living systems lies in their capacity for “observing” the motion in their surroundings, which is characterized by evolution operators. Considering a living system denoted as $(X,U_X,B_X)$ observing a quantum dynamical system $(Y,U_Y)$, I posit that X acquires all the information regarding the motion of Y, encompassing the properties of the evolution operators $U_Y$. Consequently, there exists a subsystem $X_Y$ within the system X such that $(U_Y,\circ )$ is isomorphic to $(X_Y,*)$—signifying a bijection $F:U_Y\to X_Y$ where, for all $u,v\in U_Y$, if $u\circ v\in U_Y$, then $F\left(u\right)*F\left(v\right)\in X_Y$ and $F\left(u\right)*F\left(v\right)=F(u\circ v)$. In such cases, we assert that Y is observed by X, designating the system $(X,U_X,B_X)$ as a living system.

Living organisms cannot be immortal; they are destined to experience mortality and cease functioning at a specific time. In this context, a living system X is deemed to be dead at a designated time $t_d$ if it ceases to manifest the defining characteristics of a living system for all times $t\ge t_d$.

2.3. The Core of a Living System

Cells constitute the fundamental units of a living organism, and the information within a cell is stored in ribonucleic acids such as DNAs or RNAs, which I term the “core” of the cell. Consequently, it becomes essential to establish a mathematical definition for the “core” in the context of living systems. Consider a living system denoted as $(X,U_X,B_X)$, where $C_X$ and $X^0$ function as subsets of X. Assuming $(X^0,U_X)$ is isomorphic to $(C_X,U_X)$ and there exists a subset $B_X^0\subseteq X^0$ transformed to an absorbing set $B_X\subseteq X$ by operators $P_1,\cdots ,P_n$, the subset $C_X$ is designated as the core of X. The time evolution operator $U_{t^{\prime }-t}$ acts on the core $C_X$ at time t and transforms it into the new core $C_X^{\prime }$ at time $t^{\prime }$. This temporal evolution of the core serves as the fundamental mechanism underpinning the overall evolution of living systems.

The cell has the ability to multiply or generate copies through the information encoded in its ribonucleic acids. In the context of a living system denoted as $(X,U_X,B_X)$, if there exists a subset $C_{X^{\prime }}$ of X such that $(C_{X^{\prime }},U_X)$ is isomorphic to $(C^{\prime }X,U_X)$, and $C{X}^{\prime }$ serves as the core of a living system $(X^{\prime },B_{X^{\prime }},U_{X^{\prime }})$, then we designate the living system $X^{\prime }$ as a copy of X. The operator responsible for transforming $C_X$ in the space ${\mathcal{M}}_H$ into the pair $(C_X^{\prime },C_{X^{\prime }})$ in the space ${\mathcal{M}}_H\times {\mathcal{M}}_H$ is referred to as a copy operator, symbolized by $\widehat{C}$.

3. Biological Cells

We consider a biological cell denoted as $B_X$. Let ${\mathbb{T}}_X=\{t_0,t_1,\cdots ,t_n\}$ be its lifespan, where $t_0$ is the time of its inception and $t_n$ is the time of its termination. Let X be the set of all entities (such as mass or energy quanta) absorbed by $B_X$ (or in $B_X$) at time $t_i\in {\mathbb{T}}_X$. The elements of X are represented by $(x_k,t_i)$. The time evolution operators on X are represented by

$$U_{t_j-t_i}^X(x_k,t_i)=(x_k,t_j),$$

where $t_i,t_j\in {\mathbb{T}}_X,t_j\ge t_i$ and $(x,t_i),(x,t_j)\in X$. Then $(X,U_X,B_X)$ is a quantum dynamical dissipative system, where $U_X$ is the set of time operators on X and $B_X$ can be considered as an absorbing set of X. The nucleic acids (such as DNA and RNA) inside the nucleus of the cell $B_X$ along with relevant entities in X (such as energy quanta, amino acids, phospholipids, ...), can be considered as its core.

Defining a binary operator in mathematics is straightforward, but finding a real-world analogue poses challenges. To address this, I leverage the concept of superposition states in quantum mechanics and exploit the properties of membrane proteins within cells. The structure of proteins, such as $p_0$ on the cell membrane, can be influenced by a low-strength electric field [11,12]. The presence of a membrane potential [13] has the potential to induce alterations in the structure of proteins on the membrane of $B_X$. Considering a protein on the membrane, denoted as $p_0$, I define the set $P=p_0,p_1,\cdots ,p_m$ encompassing all structures of protein $p_0$ that can be modified by membrane potential (remaining unchanged if the membrane potential is insufficient, denoted as $p_0$). This modification is contingent upon the cell being alive, ensuring $m\le n$. Assuming the protein exists in state $p_i$ at time $t_j$, where i and j are selected from the set $0,1,\cdots ,m$ with the condition that $j\ge i$, I postulate that at time $t_k\ge t_j$, the protein’s state evolves into a superposition state, a composition of both $p_j$ and $p_i$. Subsequently, at time $t_l>t_k$, the protein’s state undergoes a transition to the state $p_k$.

If we denote the state $p_j$ at time $t_j$ as $(p_i,t_j)$, we can define a binary combination (denoted by *) between two such state as follows

$$(p_i,t_j)*(p_j,t_k)=(p_i,t_k).$$

It is evident that the set $P_t$, consisting of all such state $(p_i,t_j)$, is a subset of X. The set $P_t$ with the binary combination * is isomorphic to the set $U_Y$ of time operators defined on a quantum dynamical system Y. Therefore, $(X,U_X,B_X)$ is a living system.

Let’s represent the state $p_i$ by the vector $|p_i\rangle$, for $i=1,\cdots ,m$. Suppose all the neighboring states $|p_i\rangle ,|p_{i+1}\rangle$ are orthogonal (i.e., $\langle p_i|p_{i+1}\rangle =0$), and the protein is in state $p_i$ at time instant ${\tau }_i$. Then, the set P of such states of proteins can be considered as a quantum clock [14,15,16] with a time resolution given by

$${\tau }_{i+1}-{\tau }_i=\delta {\tau }_i\ge {\displaystyle \frac{\hslash }{2\Delta E_i}},$$

where $\Delta E_i$ is the corresponding energy uncertainty. This implies that quantum clocks can exist in every living cell containing membrane proteins that can change states over time.

Numerous individual living systems $(X_i,U_{x_i},B_{X_i})$ can combine to form a living system $(X,U_X,B_X)$, where $X={\bigcup }_i{X}_i,U_X={\bigcup }_i{U}_{X_i},B_X={\bigcup }_i{B}_{X_i}$.

If the protein cannot return to the initial state $p_0$, we say that it has undergone degeneration. If the proteins within the cell membrane cannot enter a superposition state, the cell wouldn’t observe any quantum dynamical system and thus cannot be considered as a living system. Without the membrane potential, the proteins would remain unaffected by the electric fields, preventing necessary structural changes for entering a superposition state. The presence of an environment conducive to membrane potential, facilitated by water, becomes essential. This elucidates why living systems need water.

4. Conclusions

In conclusion, while this mathematical model of life may not comprehensively encapsulate all the intricacies of living organisms, it stands as a vital link that unites fundamental physics theories with biology. Significantly, it enables the distinction between living and non-living systems. While certain facets are presently confined to theoretical descriptions, their validation through experimentation is imperative, particularly concerning the superposition state of membrane proteins influenced by a low-strength electric field. This model lays the groundwork for further exploration and empirical validation, fostering a deeper understanding of the complex dynamics inherent in living systems.