Formulating a Mathematical Model for Living Systems

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Formulating a Mathematical Model for Living Systems

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Toan Nguyen^*

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Submitted:

07 October 2024

Posted:

07 October 2024

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Abstract

Prigogine’s 1978 concept of dissipative structures, drawing parallels with living systems, forms the basis for exploring life’s unique traits. However, these identified similarities prove insufficient in capturing the entirety of life. To address this gap, our proposed modeling approach emphasizes the distinctive ability of living organisms to observe other systems—an attribute intricately tied to quantum mechanics’ "measurement" processes, as highlighted by Howard Pattee. This article introduces a comprehensive mathematical model centered on quantum dynamical dissipative systems, portraying living systems as entities defined by their observational capacities within this framework. The exploration extends to the core dynamics of these systems and the intricacies of biological cells, including the impact of membrane potentials on protein states. Within this theoretical structure, the model is expanded to multicellular living systems, revealing how cells observe quantum dynamical systems through protein state changes influenced by membrane potentials. The conclusion acknowledges the current theoretical status of the model, underscoring the crucial need for experimental validation, particularly regarding the superposition state of membrane proteins under the influence of an electric field.

Keywords:

Subject: Biology and Life Sciences - Biophysics

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

M

to be a Grothendieck universe, defined as the universe comprising only ’pure sets’ [7]. In this context,

H

represents the set of all linear maps from

M

R

, and

I \subseteq R

is a finite set consisting of specific (finite) values

h (x)

for some

x \in M

and certain linear maps

h \in 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

M_{H} \subseteq M

be a Hilbert space equipped with a norm

∥x∥ = \sqrt{〈 x, x 〉}

and a distance function

d (x, y) = ∥x - y∥

defined for each

x, y \in M_{H}

. Then

M_{H}

is also a Banach space, and the set of all (Lipschitz) continuous linear maps from

M_{H}

R

is the dual space of

M_{H}

(denoted by

M_{H}^{*}

M_{H}^{*}

is also a Hilbert space [8,9].

Let

I_{H} \subseteq I

be a set of finite eigenvalues of operators on

M_{H}

, then

I_{H}

is finite. Let

X \subseteq M_{H}

be a closed subset of

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 (y) = 〈 x, y 〉

for all

y \in X

. We denote f as

〈 x |

, y as

| y 〉

, and

〈 x, y 〉

〈 x | y 〉

. Additionally, we define a set

U_{X}

of evolution operators represented by

\begin{matrix} U_{t - t_{0}} : X & \to X, \\ | x_{t_{0}} 〉 & \mapsto | x_{t} 〉, \end{matrix}

where

t, t_{0} \in T_{X} \subseteq I_{X}

and

t \geq t_{0}

. The operators in

U_{X}

satisfy the following properties:

U_{t - t} | x 〉 = U_{0} | x 〉 = | x 〉, U_{t - t_{1}} \circ U_{t_{1} - t_{0}} | x 〉 = U_{t - t_{0}} | x 〉

, where ∘ is the composition of operators. These operators are known as the time-evolutions or the propagators [9]. Each element

U_{t, t_{0}}

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 T_{X}

such that

U_{t - t_{0}} (D) \subseteq B_{X}

for all

t, t_{0} \in I_{X}

with

t \geq t_{D} \geq t_{0}

, where

U_{t - t_{0}} (D) = {U_{t - t_{0}} | x 〉| 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_{i j}

, where

i, j \in Z_{+}

and

i \leq j

. Let * is a binary operator defined on

X_{Y}

such that

x_{i j} * x_{j k} = x_{i k}

whenever

x_{i j}, x_{j k}, x_{i k}

are in

X_{Y}

. The set

X_{Y}

is not necessarily closed under *, as there may exist

x_{a b}, x_{b c}

X_{Y}

such that

x_{a c}

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 (u) * F (v) \in X_{Y}

and

F (u) * F (v) = 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 \geq 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}, \dots, P_{n}

, the subset

C_{X}

is designated as the core of X. The time evolution operator

U_{t^{'} - t}

acts on the core

C_{X}

at time t and transforms it into the new core

C_{X}^{'}

at time

t^{'}

. 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^{'}}

of X such that

(C_{X^{'}}, U_{X})

is isomorphic to

(C^{'} X, U_{X})

, and

C X^{'}

serves as the core of a living system

(X^{'}, B_{X^{'}}, U_{X^{'}})

, then we designate the living system

X^{'}

as a copy of X. The operator responsible for transforming

C_{X}

in the space

M_{H}

into the pair

(C_{X}^{'}, C_{X^{'}})

in the space

M_{H} \times M_{H}

is referred to as a copy operator, symbolized by

\hat{C}

3. Biological Cells

We consider a biological cell denoted as

B_{X}

. Let

T_{X} = {t_{0}, t_{1}, \dots, 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 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 T_{X}, t_{j} \geq 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}, \dots, 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 \leq n

. Assuming the protein exists in state

p_{i}

at time

t_{j}

, where i and j are selected from the set

0, 1, \dots, m

with the condition that

j \geq i

, I postulate that at time

t_{k} \geq 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}

(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} 〉

, for

i = 1, \dots, m

. Suppose all the neighboring states

| p_{i} 〉, | p_{i + 1} 〉

are orthogonal (i.e.,

〈 p_{i} | p_{i + 1} 〉 = 0

), and the protein is in state

p_{i}

at time instant

τ_{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

τ_{i + 1} - τ_{i} = δ τ_{i} \geq \frac{ℏ}{2 Δ E_{i}},

where

Δ 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 = ⋃_{i} X_{i}, U_{X} = ⋃_{i} U_{X_{i}}, B_{X} = ⋃_{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.

Acknowledgments

I express my sincere gratitude to my parents and my wife for their unwavering support. Special thanks go to my younger brother, Nguyen Dac Khoi Nguyen, for his invaluable contributions through critical discussions and assistance in discovering pertinent mathematical textbooks.

References

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Formulating a Mathematical Model for Living Systems

Abstract

1. Introduction

2. Mathematical Model

2.1. Quantum Dynamical Dissipative Systems

2.2. Living Systems

2.3. The Core of a Living System

3. Biological Cells

4. Conclusions

Acknowledgments

References

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