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.
> 10 000 articles
The human body is a complex network of networks.
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.
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
The regulation of gene expression involves a complex network of interacting elements. Expression profiles of different tissues and cells can be a powerful tool in aging research in the context of the dynamics of the gene regulatory network. In such a graph, the determining participants are complexes of various transcription factors that regulate gene expression.
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).
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).
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.
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).
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.
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.
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).
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.
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.
An example of a Preferential Attachment in the evolution of the network in the Barabasi-Albert model. Connections are more likely to form between nodes with a large valence.
>>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.
Real specrtum for co-expression and complex for
gene regulation.
>>Conservation laws.
>>Evolution 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.
Clique percolation method & Preferential attachement
This parameters reflect the type of gene network dynamics: chaotic, deterministic or critical.
R-statistics, Inverse Participation Ratio (IPR).
We believe it is possible to develop a rigorous and comprehensive aging model based on the analysis of GRN dynamics and its spectral properties.
We believe it is possible to develop a rigorous and comprehensive aging model based on the analysis of GRN dynamics and its spectral properties.
Spectral analysis was successfully used for connectome studies and it produced several outstanding results:
Complicated dynamics of connectome can be described in terms of relatively simple models whose predictions can be experimentally verified.
>>>>>>>
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.
Questions that should be answered with network theory to radically extend human lifespan.
>>
What does the extremely long lifespan of some animals mean within the framework of network theory?
>>
Biomarker of aging for resilient organisms.
>>
How to select combinatorial therapy against aging?
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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