# Waiting for a Mathematical Theory of Living Systems from a Critical Review to Research Perspectives

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## Abstract

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## 1. Motivations and Plan of the Paper

What is life and how science can contribute to a mathematical theory of living systems?

## 2. From a Philosophical-Mathematical Excursus towards a Mathematics of Life

#### 2.1. A Philosophical–Mathematical Excursus

We are surrounded by complex systems. Familiar examples include power grids, transportation systems, financial markets, the Internet, and structures underlying everything from the environment to the cells in our bodies. Mathematical models can guide us in understanding these systems […].

Living matter, while not eluding the “laws of physics” as established up to date, is likely to involve “other laws of physic” hitherto unknown, which however, once they have been revealed, will form just as integral a part of science as the former.

Although living systems obey the laws of physics and chemistry, the notion of function or purpose differentiates biology from other natural sciences. Organisms exist to reproduce, whereas, outside religious belief, rocks and stars have no purpose. Selection for function has produced the living cell, with a unique set of properties that distinguish it from inanimate systems of interacting molecules.

#### 2.2. On a Strategy towards a Mathematics of Living Systems

- Expression of functions: Living entities, which may be defined micro-systems to account that their representation is at the lower scale with respect to that of the global system. An alternative definition is active particles, shortly a-particles, see [18]. These have the ability to express a function called activity. These entities are able to develop specific strategies and organization abilities that depend on the state of the surrounding entities and environment.
- Heterogeneity: The ability to express the activity is not the same for all micro-systems as the expression of heterogeneous behaviors is a common feature of a great part of living systems.
- Nonlinearity of interactions: Interactions are nonlinearly additive, nonlocal, as they may involve entities that are not immediate neighbors and, in some cases, asymmetric. Active particles are sensitive not only to a selection of other individual entities, but to the overall system as a whole.
- Learning ability: Living systems receive inputs from the environments and have the ability to learn from past experience. Accordingly, the rules of interaction and activity they develop evolves in time.
- Functional subsystems: Active particles can aggregate into groups, called functional subsystems, shortly FSs, where they pursue the same objectives, share the same activity and interaction rules.
- Output of interactions: Interactions modify the activity and generate proliferative and destructive events.
- Darwinian mutations and selection: All living systems are evolutionary, as interactions can generate, by birth of aggregations, new entities that are increasingly fitted to the environment, that, in turn, generate new entities again more fitted to the environment.

- The overall system of a-particles is subdivided into functional subsystems accounting for the activity variables rather than physical properties of the matter of the system.
- The state of each FS is described by the one particle distribution function over the micro-state of the a-particles belonging to it: ${f}_{i}={f}_{i}(t,\mathit{x},\mathit{v},\mathbf{u})$ with ${f}_{i}\ge 0$ for all $t,\mathit{x},\mathit{v},\mathbf{u}$, where t is time, $\mathit{x}$ and $\mathit{v}$ model the localization and velocity of the a-particle, and $\mathbf{u}$ is the activity shared by all a-particles of the FS.
- Derivation of a general mathematical structure with the aim of offering the conceptual framework toward the derivation of specific models. This structure is required to express the dynamics of the complexity features of living systems.
- Derivation of specific models corresponding to well-defined classes of systems by implementing the said structure with suitable models of individual-based, micro-scale, interactions.

- The phenomenological interpretation of living systems can take advantage of specific measurement devices, but it is always somehow conditioned by individual sensitivity within the framework of the sensitive world by Immanuel Kant [23,24] or even within the framework of the artificial world by Herbert Simon [25].
- Generally, the derivation of models is developed by a hierarchy in which each micro-system firstly learn from the other micro-systems and subsequently, modifies the mechanical state.
- The application of models generates challenging analytic and computational problems which require new inventions to tackle them in real applications by the critical analysis developed in the next section after a review of specific classes of models.

## 3. From Active Particles Methods to Selected Applications

#### 3.1. Mathematical Models in Space Homogeneity

- Interaction rate for conservative dynamics:${\eta}_{ik}({\mathbf{u}}_{*},{\mathbf{u}}^{*})$, which models the frequency of the interactions between a candidate i-micro-system with state ${\mathbf{u}}_{*}$ and a field k-micro-system with state ${\mathbf{u}}^{*}$, where the label i denotes the FS.
- Interaction rate for non-conservative dynamics:${\mu}_{ik}({\mathbf{u}}_{*},{\mathbf{u}}^{*})$, is analogous to ${\eta}_{ik}$, but corresponding to proliferative and destructive interactions.
- Transition probability density:${\mathcal{C}}_{ik}\left[\mathbf{f}\right]({\mathbf{u}}_{*}\to \mathbf{u}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{\mathbf{u}}_{*},{\mathbf{u}}^{*})$, which denotes the probability density that a candidate i-micro-system, with state ${\mathbf{u}}_{*}$, ends up into the state of the test micro-system of the same FS after an interaction with a field k-micro-system.
- Proliferative term:${\mathcal{P}}_{ik}\left[\mathbf{f}\right]({\mathbf{u}}_{*}\to \mathbf{u}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{\mathbf{u}}_{*},{\mathbf{u}}^{*})$, which models the proliferative events for a candidate h-particle, with state ${\mathbf{u}}_{*}$, into the same FS after interaction with a field k-particle with state ${\mathbf{u}}^{*}$.
- Destructive term:${\mathcal{D}}_{ik}\left[\mathbf{f}\right](\mathbf{u},{\mathbf{u}}^{*})$, which models the rate of destruction for a test i-particle in its own functional subsystem after an interaction with a field k-particle with state ${\mathbf{u}}^{*}$.

- Modeling the competition between a virus and the immune system, accounting for the progression of the virus and the activation of the immune system has been proposed in [32].
- Somehow inspired to the above literature is the study developed in [33] to model the in-host dynamics in the upper respiratory tracts, where the virus proliferates, although contrasted by the immune system attempting to reduce the invasion ability of the virus. Further developments have been presented in [34] to consider the role of mutations and onset of new variants.

#### 3.2. Mathematical Models with Space Dynamics

#### 3.3. A Forward Look to Perspectives in Modeling

- Vector activity variable and related hierarchy of the development of the dynamics. Indeed, more that one variable might be necessary to characterize micro-systems. The difficulty in identifying the correlations between them suggests to understand whether the decision-making process, as often happens in living systems, follows a sequence linked to the components of the activity.
- Correlation of the role of the activity to the dynamics of the mechanical variables. This means selecting the velocity directions and then, adapting the speed to the local flow conditions. This selection is activity-driven. As an example, increasing stress conditions increases the tendency towards the main (overcrowded) stream rather than the seeking less crowded areas.
- Including mutations and selection also in the study of social systems. This type of dynamics is typical of biological systems. However, a dynamic across functional subsystems can be observed also in social systems. For instance in opinion dynamics micro-systems can move from a group of interest to another one.
- Exploring new mathematical structures. The reasoning on the possible developments of the mathematical structures reviewed in this section is definitely a strategic objective. However, a global vision should also consider other frameworks such as those mentioned in Section 2.2, i.e., behavioral swarms [19] and Boltzmann and Fokker-Plank methods [5] and even possible interactions between different frameworks.
- Exploring multi scale methods. The Hilbert’s sixth problem concerns the search of a unified mathematical approach to physical theories. A conceivable preliminary step would be the derivation of models at all scales by the same principles and, subsequently, the passage to mathematical (hence rigorous) methods from the low to the higher scale. This passage is depicted in Figure 2.

## 4. On a Forward Look to a Mathematical Theory of Living Systems

After selecting of a specific class of case studies, databases can be organized to collect data on a selected number of interactions which are specific to the case studies. The selection should correspond to the causality action characterizing each interaction. Therefore, selecting in the database the dynamics close, with a suitable metric, to those treated by a model would lead to the calibration of the model.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Burini, D.; Chouhad, N.; Bellomo, N.
Waiting for a Mathematical Theory of Living Systems from a Critical Review to Research Perspectives. *Symmetry* **2023**, *15*, 351.
https://doi.org/10.3390/sym15020351

**AMA Style**

Burini D, Chouhad N, Bellomo N.
Waiting for a Mathematical Theory of Living Systems from a Critical Review to Research Perspectives. *Symmetry*. 2023; 15(2):351.
https://doi.org/10.3390/sym15020351

**Chicago/Turabian Style**

Burini, Diletta, Nadia Chouhad, and Nicola Bellomo.
2023. "Waiting for a Mathematical Theory of Living Systems from a Critical Review to Research Perspectives" *Symmetry* 15, no. 2: 351.
https://doi.org/10.3390/sym15020351