Abstract
Symmetries are ubiquitous in nature. Almost all organisms have some kind of “symmetry”, meaning that their shape does not change under some geometric transformation. This geometrical concept of symmetry is intuitive and easy to recognize. On the other hand, the behavior of many biological systems over time can be described with ordinary differential equations. These dynamic models may also possess “symmetries”, meaning that the time courses of some variables remain invariant under certain transformations. Unlike the previously mentioned symmetries, the ones present in dynamic models are not geometric, but infinitesimal transformations. These mathematical symmetries can be related to certain features of the system’s dynamic behavior, such as robustness or adaptation capabilities. However, they can also arise from questionable modeling choices, which may lead to non-identifiability and non-observability. This paper provides an overview of the types of symmetries that appear in dynamic models, the mathematical tools available for their analyses, the ways in which they are related to system properties, and the implications for biological modeling.
Highlights
Symmetrical features are abundant in the biological world
Despite the fact that the distinction made in (1) between states, parameters, and inputs is the usual practice in dynamic modeling, we will group them together in an augmented state vector x ∈ Rn with dimension n = nx + nu + nw + nθ
Lie symmetries can be found by solving a system of differential equations, whose solutions are the expressions of the transformations admitted by the model
Summary
Symmetrical features are abundant in the biological world. Practically all organisms have some kind of symmetry in three-dimensional space. It presents the fundamental concepts of Lie symmetries and related mathematical tools (Section 2) It describes a number of features found in dynamic models of biological systems, and their connections to symmetries, as well as the biological and modeling implications (Section 3). Despite the fact that the distinction made in (1) between states, parameters, and inputs is the usual practice in dynamic modeling, we will group them together in an augmented state vector x ∈ Rn with dimension n = nx + nu + nw + nθ The reason for this is that all those variables can be involved in the symmetries that we consider.
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