Mathematical Physics for Discoveries in Aging

Using the developments of the cutting edge of mathematical physics, we study the aging of various organisms, analyzing it as the dynamics of a complex network system

Aging is a process that manifests itself at all levels of organization in a living organism.

López-Otín, Carlos, et al. "The hallmarks of aging." Cell 153.6 (2013): 1194-1217

Understanding the key molecular factors that are the drivers of the reorganization of the human body over time would allow us to develop methods of influencing them and thereby create a therapy for old age.

More than ten thousand articles on aging are published yearly. However, there has been no significant progress in prolonging human life so far.

This situation can be changed by models that use state-of-the-art mathematical physics methods and are built on experimental biological data.

This situation can be changed by models that use state-of-the-art mathematical physics methods and are built on experimental biological data.

> 10 000 articles

Conceptually, a human (or any other) organism can be considered as an evolving complex network.

In such a network, or a graph,**vertices** represent distinct physiological units (PUs) - for instance, genes, proteins, methylation sites, etc., and **edges** represent time-dependent interactions between them. For each **vertex** there is a **degree of freedom** - a value that changes specifically to given network due to interaction with other vertices.

In such a network, or a graph,

Lehnertz, Klaus, Timo Bröhl, and Thorsten Rings. "The human organism as an integrated interaction network: recent conceptual and methodological challenges." Frontiers in Physiology (2020): 1694

Gene network.

Two types of gene networks that we analyze.

Gene coexpression network (GCN)

1

GCN is an undirected graph in which an edge between genes means coexpression. That is, the expression of two genes is correlated, but one of the genes does not necessarily activates or represses the other.

Genes x, y, z are coexpressed, but regulation rules are unknown.

Gene-regulatory network (GRN)

2

GRN is an oriented and weighted graph, in which for a pair of genes it is clear, which of them is a transcription factor – activator or repressor, and which is the target gene.

For such coexpression of x, y, z there might be many types of topology of regulatory network:

Equations that describe the dynamics of aging for interacting organism's physiological units.

Turchetti, Giorgio, Fabio Luciani, and Luca Mariani. "Environmental randomness and survival probability." PLVC e. The MIRIAM Project Series, editor, Mathematical Modeling in Biology and medicine, 5th ESMTB Conference. 2003.

Mariani, L., G. Turchetti, and C. Franceschi. "Chronic antigenic stress, immunosenescence and human survivorship over the 3 last centuries: heuristic value of a mathematical model." Mechanisms of aging and development 124.4 (2003): 453-458.

Podolskiy, Dmitriy, et al. "Critical dynamics of gene networks is a mechanism behind aging and Gompertz law." arXiv preprint arXiv:1502.04307 (2015).

Mariani, L., G. Turchetti, and C. Franceschi. "Chronic antigenic stress, immunosenescence and human survivorship over the 3 last centuries: heuristic value of a mathematical model." Mechanisms of aging and development 124.4 (2003): 453-458.

Podolskiy, Dmitriy, et al. "Critical dynamics of gene networks is a mechanism behind aging and Gompertz law." arXiv preprint arXiv:1502.04307 (2015).

For interacting degrees of freedom, or **physiological units** of the organism - genes, metabolites, immune system, etc., the **dynamics in the process of aging** can be described by the following differential equation:

For **short-lived organisms**, aging can be described by the dynamics of a single variable following the **Langevin equation**, meaning that such variable can be a **biomarker of aging**:

Mariani, L., G. Turchetti, and C. Franceschi. "Chronic antigenic stress, immunosenescence and human survivorship over the 3 last centuries: heuristic value of a mathematical model." Mechanisms of aging and development 124.4 (2003): 453-458.

"Podolskiy, Dmitriy, et al. "Critical dynamics of gene networks is a mechanism behind aging and Gompertz law." arXiv preprint arXiv:1502.04307 (2015)."

Tarkhov, Andrei E., Kirill A. Denisov, and Peter O. Fedichev. "Aging clocks, entropy, and the limits of age-reversal." bioRxiv (2022).

"Podolskiy, Dmitriy, et al. "Critical dynamics of gene networks is a mechanism behind aging and Gompertz law." arXiv preprint arXiv:1502.04307 (2015)."

Tarkhov, Andrei E., Kirill A. Denisov, and Peter O. Fedichev. "Aging clocks, entropy, and the limits of age-reversal." bioRxiv (2022).

However, for human and long-lived organisms, such behavior takes place only at a specific moment, namely, at the end of life or in the case of several chronic diseases.

vector fields associated with average environmental stress and the signalling or control processes in the body, which are considered as changing slowly

random rapidly fluctuating force

negative constant, up to multiplication the smallest (largest negative) eigenvalue of the matrix A(t)

slowly changing matrix describing the interaction between physiological units

vector with components corresponding to the values of interacting physiological units, for example, gene expression levels and concentrations of metabolites

Universality among different physiological units.

The beauty of this theory consists in the **invariance of alpha** for a particular species across various types of data (alpha does not depend on the selection of physiological units).

This means that the**universal behavior** (inherent in systems operating in near-critical mode) is taking place: independently of the microscopic structure, the behavior of two systems differing in the choice of so-called physiological units is described by the same quantities.

This means that the

Equations for a limited number of degrees of freedom.

In a biological system, **interactions of degrees of freedom** are highly complex. Such a system is a **thermodinamically large network** for which it is impossible to take into account all interactions.

Therefore, we can only consider the dynamics of a subset of degrees of freedom or physiological units, the so-called**"truncated" models**. Nevertheless, even this approach gives good results, which indicates that the dropped degrees of freedom do not lead to a strong distortion of the overall picture.

Podolskiy, Dmitriy, et al. "Critical dynamics of gene networks is a mechanism behind aging and Gompertz law." arXiv preprint arXiv:1502.04307 (2015).

Avchaciov, Konstantin, et al. "Identification of a blood test-based biomarker of aging through deep learning of aging trajectories in large phenotypic datasets of mice." bioRxiv (2020).

Therefore, we can only consider the dynamics of a subset of degrees of freedom or physiological units, the so-called

Podolskiy, Dmitriy, et al. "Critical dynamics of gene networks is a mechanism behind aging and Gompertz law." arXiv preprint arXiv:1502.04307 (2015).

Avchaciov, Konstantin, et al. "Identification of a blood test-based biomarker of aging through deep learning of aging trajectories in large phenotypic datasets of mice." bioRxiv (2020).

Criticality.

The regulatory network of the organism (and, in particular, the gene-regulatory network), as a dynamic system, functions near the **bifurcation ****point**.

Balleza, Enrique, et al. "Critical dynamics in genetic regulatory networks: examples from four kingdoms."*PLoS One* 3.6 (2008): e2456.

Kim, Hyobin, and Hiroki Sayama. "The role of criticality of gene regulatory networks in morphogenesis."*IEEE Transactions on Cognitive and Developmental Systems* 12.3 (2018): 390-400.

Kim, Hyobin, and Hiroki Sayama. "How criticality of gene regulatory networks affects the resulting morphogenesis under genetic perturbations."*Artificial Life* 24.02 (2018): 85-105.

Valverde, Sergi, et al. "Structural determinants of criticality in biological networks."*Frontiers in physiology* 6 (2015): 127.

Balleza, Enrique, et al. "Critical dynamics in genetic regulatory networks: examples from four kingdoms."

Kim, Hyobin, and Hiroki Sayama. "The role of criticality of gene regulatory networks in morphogenesis."

Kim, Hyobin, and Hiroki Sayama. "How criticality of gene regulatory networks affects the resulting morphogenesis under genetic perturbations."

Valverde, Sergi, et al. "Structural determinants of criticality in biological networks."

Universality among different physiological units.

This means that the universal behavior (inherent in systems operating in near-critical mode) is taking place: independently of the microscopic structure, the behavior of two systems differing in the choice of so-called physiological units is described by the same quantities.

Equations for a limited number of degrees of freedom.

In a biological system, interactions of degrees of freedom are highly complex. Such a system is a thermodinamically large network for which it is impossible to take into account all interactions.

Therefore, we can only consider the dynamics of a subset of degrees of freedom or physiological units, the so-called "truncated" models. Nevertheless, even this approach gives good results, which indicates that the dropped degrees of freedom do not lead to a strong distortion of the overall picture.

Podolskiy, Dmitriy, et al. "Critical dynamics of gene networks is a mechanism behind aging and Gompertz law." arXiv preprint arXiv:1502.04307 (2015).

Avchaciov, Konstantin, et al. "Identification of a blood test-based biomarker of aging through deep learning of aging trajectories in large phenotypic datasets of mice." bioRxiv (2020).

Therefore, we can only consider the dynamics of a subset of degrees of freedom or physiological units, the so-called "truncated" models. Nevertheless, even this approach gives good results, which indicates that the dropped degrees of freedom do not lead to a strong distortion of the overall picture.

Podolskiy, Dmitriy, et al. "Critical dynamics of gene networks is a mechanism behind aging and Gompertz law." arXiv preprint arXiv:1502.04307 (2015).

Avchaciov, Konstantin, et al. "Identification of a blood test-based biomarker of aging through deep learning of aging trajectories in large phenotypic datasets of mice." bioRxiv (2020).

Criticality.

The regulatory network of the organism (and, in particular, the gene-regulatory network), as a dynamic system, functions near the bifurcation point.

Balleza, Enrique, et al. "Critical dynamics in genetic regulatory networks: examples from four kingdoms."*PLoS One* 3.6 (2008): e2456.

Kim, Hyobin, and Hiroki Sayama. "The role of criticality of gene regulatory networks in morphogenesis."*IEEE Transactions on Cognitive and Developmental Systems* 12.3 (2018): 390-400.

Kim, Hyobin, and Hiroki Sayama. "How criticality of gene regulatory networks affects the resulting morphogenesis under genetic perturbations."*Artificial Life* 24.02 (2018): 85-105.

Valverde, Sergi, et al. "Structural determinants of criticality in biological networks."*Frontiers in physiology* 6 (2015): 127.

Balleza, Enrique, et al. "Critical dynamics in genetic regulatory networks: examples from four kingdoms."

Kim, Hyobin, and Hiroki Sayama. "The role of criticality of gene regulatory networks in morphogenesis."

Kim, Hyobin, and Hiroki Sayama. "How criticality of gene regulatory networks affects the resulting morphogenesis under genetic perturbations."

Valverde, Sergi, et al. "Structural determinants of criticality in biological networks."

Analysis of **gene-gene connections** changes over time allow to determine **network invariants**, or conservation laws, for particular species during aging.

Analysis of **matrix eigenvalues** and **level spacing distribution** give us a representation of gene network structure.

>>Spectral properties of gene networks.

>>Renormalization group method (RN approach).

This method has found wide application in statistical physics. It allows to explore the **neighborhood of a critical point** and extract information about critical indexes, a set of values that exhaustively describes the statistical properties of a system near a critical point.

One of the key results of previous works is the **correlator of time fluctuations**. The analysis of the properties of this correlator is very similar to the classical approach of the renormalization group for phase transitions.

Podolskiy, Dmitriy, et al. "Critical dynamics of gene networks is a mechanism behind aging and Gompertz law." arXiv preprint arXiv:1502.04307 (2015).

Methods for the analysis of the properties of networks of interacting physiological units of the body and their influence on the dynamics of aging.

gene regulation.

>>Conservation laws.

>>E**volution** of gene networks.

This methods allow to identify **laws of evolution** of gene network.

An evolution of matrix structure in time can indirectly**describe aging**.

An evolution of matrix structure in time can indirectly

Clique percolation method & Preferential attachement

This parameters reflect the **type** of gene network **dynamics**: chaotic, deterministic or critical.

R-statistics, Inverse Participation Ratio (IPR).

Spectral analysis was successfully used for connectome studies and it produced several outstanding results:

>>>>>>>

The **renormalization group** method has already found application in a wide range of **condensed matter physics** problems (for example, the analysis of phase transitions from a magnetic phase to a non-magnetic one) and **soft matter physics** (various phase transitions in polymers, including peptides). The ideas of the renormalization group approach go back to **dynamic systems**, which strengthens our expectation that this approach will be useful for aging models.

Ódor, Géza, and Jeffrey Kelling. "Critical synchronization dynamics of the Kuramoto model on connectome and small world graphs." Scientific reports 9.1 (2019): 1-10.

It was shown that the **human connectome** operates near **criticality**, and properties of this network can be captured by the quite simple mathematical model (**Kuramoto model**).

Anokhin, K., et al. "Spectral peculiarity and criticality of the human connectome." arXiv preprint arXiv:1812.06317 (2018).

The comparative analysis, based on **structural connectomes** for several organisms - **C. elegans**, **macaque**, and **human**, has demonstrated that the human connectome differs from the other connectomes. This **difference** can be described quantitatively by **graph spectra**.

Groups of genes that determine parameters of stable states and transitions between stable states of long-lived organisms.

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The influence of the appearance of **another level of organization** (organs) on the aging process and on the changes of interaction laws.

Anderson, Philip W. "More is different: broken symmetry and the nature of the hierarchical structure of science." Science 177.4047 (1972): 393-396.

In **particle physics** the appearance of another level of an organization significantly affects the properties, internal interactions, and dynamics of the system.

The **arctic sea sponge** (*Anoxycalyx joubini*), belongs to the animal kingdom and lives about **15,000 years**. The peculiarity of the sponge is the **absence of actual tissues and organs**; different functions in its organism are performed by various individual cells and cellular layers.

Perhaps such a huge lifespan of a sponge is due to the **specific** **conservation laws** of its gene-regulatory network. In addition, aging can be associated with the **formation of structures** in the network, and organisms with **low clusterization** live longer.

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Is it possible to **change** the sign of the smallest eigenvalue of matrix A by any intervention, thereby transforming an **unstable** organism **into a stable** one?

Is it possible to achieve a **synergistic effect** by selecting a group of drugs that changes **distinct eigenvalues** of matrix?

How to apply **anti-aging interventions** correctly **in time** - is pulse therapy the better strategy?

How to theoretically predict changes in the **properties of matrix A** in response to gene therapy, drug therapy, or other interventions?

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