Frontiers is organizing a series of Research Topics to highlight the latest advancements in research across the field of Applied Mathematics and Statistics. This particular editorial initiative is focused on new insights, novel developments, current challenges, latest discoveries, recent advances, and future perspectives in the field of Mathematical Biology. The Research Topic solicits brief, forward-looking contributions that describe the state of the art, outlining recent developments and major accomplishments that have been achieved and that need to occur to move the field forward. Authors are encouraged to identify the greatest challenges in the sub-disciplines, and how to address those challenges. The goal of this special edition Research Topic is to shed light on the progress made in the past decade in the Mathematical Biology field, and on its future challenges to provide a thorough overview of the field. This article collection will inspire, inform and provide direction and guidance to researchers in the field.

University of Franche-Comté

Discover advances and major successes achieved in the Mathematical Biology field to move it forward.

. Research in the field of cellular health has revealed two models of cell cycle progression, which use structural modeling conventions to explore cell-cycle dynamics. Farmers and governments are working together to maximize food production in a sustainable manner. An agent-based model has been developed to analyze the effects of farming practices on the environment and how different levels of cooperation between farmers can incentivize producers to maximize yields while minimizing the environmental impact. A mathematical model based on finite element methods and machine learning has been developed to optimize treatment for severe burn injuries. The model showed a high accuracy and a tremendous speedup of 1815000X, and has been applied in an online medical app that takes the patient's age into account.

Frontiers in Applied Mathematics and Statistics

Mathematical Biology

Insights in Mathematical Biology 2022

This Insights in Mathematical Biology Research Topic aimed to showcase some new developments and future research perspectives in the mathematical biology field. The three articles published in this topic are a very small sample of some of the current interests in mathematical biology, from the optimization of food production to the optimization of numerical simulations for complex mathematical models describing medical problems and the understanding of multi-scale molecular&#x2013;cellular aspects in health and disease. These articles also emphasize some of the current approaches in mathematical biology that oppose single-scale vs. multi-scale models, simple vs. complex models, and fast vs. slow numerical simulations.In the following, we briefly summarize these three papers.<a href="https://doi.org/10.3389/fams.2023.1058273">O'Hare</a> focused on maximizing food production in a sustainable manner. To investigate the competitive behavior between two different types of food producers (one where farmers harvest and make profit at the end of the season and a second one where farmers harvest and make profit continuously throughout the season), the author considered a simple individual-based framework. This framework was applied to 25 farms distributed on a spatial grid, which could interact with neighboring farms and produce food using either of the two food-producing systems. The interaction rule between the different farms was given by a simple algebraic equation that considered the distance between farms (which could belong to different farm networks: square grid, linear network, circular network, and random network) and the time since the farmers started to implement actions that could impact neighboring farms. The profit obtained by each farmer (calculated using an integral equation) depended on the running cost of each farmer to produce their crops, the price at which the crop is sold, and on the interaction rule between the different farms. The solution of this integral equation showed that the profit obtained through farming was calculated for all farms, and the results showed that profit depended on the farm's position in the landscape. Moreover, neighboring farms that produced similar crops could benefit from others' treatment of their crops, while neighboring farms that produced different crops had a negative impact on each other. The study also modeled the impact of connections between different farms (connections by a road or a river) and investigated the level of landscape pollution at various farms and how this was transported toward other connected farms. The toy individual-based model developed in this study to calculate the profit for each farmer (with its farm belonging to a spatially distributed network of farms) showed that even simple mathematical models could be used to generate new testable hypotheses for complex ecological problems. However, such toy models will have to be modified in the future to describe specific landscape conditions, specific food producers, specific pollutants (pesticides) use, and their dispersion characteristics to neighboring farms.<a href="https://doi.org/10.3389/fams.2023.1098242">Egberts et al.</a> focused on a different type of optimization: the optimization of numerical simulations in the context of a very complex mathematical model for wound healing. The model, described by nine non-linear partial differential equations, focused on the spatio-temporal interactions between different cell types (fibroblasts, myofibroblasts), collagen, and chemicals during post-burn tissue contraction. To improve the performance of the finite element numerical simulations of this 2D spatial model for cell- and tissue-scale dynamics of wound healing, the authors considered a feed-forward neural network (with two hidden layers) trained and then tested on 2D finite element simulations. The resulting optimized network was used to predict the wound boundary contraction and the total strain energy density. It was also suggested that the fast numerical simulations could be eventually integrated into medical practice through the development of apps that consider age-related parameters to predict wound contractures.<a href="https://doi.org/10.3389/fams.2023.1090753">Hodgkinson et al.</a> focused on a biological aspect underlying many medical problems: the molecular and cell-scale dynamics behind the regulation of cell cycle via a family of proteins called cyclin-dependent kinases and the p53 gene. The study presented two mathematical models described by integro-partial differential equations: the first one focused on the quantity of cyclins across cell cycle at the level of one single cell, and the second one focused on the role of cyclin, the p53 protein, and the mdm2 protein (which degrades p53) in cell-cycle dynamics at the level of whole-cell populations. In the first model, the variables describing the amount of S-cyclin and M-cyclin in a cell (where the S-cyclins and M-cyclins denoted the cyclins synthesized in the S and M phases of cell cycle) depended on time and space. In the second model, the variables describing the densities of cells in the various phases of cell cycle (S, G, M) depended on time, cyclin level synthesized in various phases, cell damage level, and p53 and mdm2 levels. Numerical simulations (using a finite difference scheme together with a mid-point integration method for integrals) with the first model showed oscillatory dynamics between the S-cyclin and M-cyclin states. Numerical simulations with the second model (using a finite difference scheme based on a predictor–corrector MacCormack scheme for time integration and central differences for the spatial gradients) showed the distributions of cell populations in S and M phases across the (cyclin, damage) space and the distribution of the cell population in the G phase across the (cyclin, damage) space, the (p53, damage) space, and the (p53, mdm2) space for various parameter values. Due to the complexity of this second model in [2] and the computational limitations on the number of dimensions that could be simulated simultaneously at a resolution that could allow enough accuracy, this second model did not include any spatial cell structure. This is an open problem that will have to be addressed in the future.In summary, the articles in this Research Topic highlight some of the current open problems in cell biology, medicine, and ecology. All these articles also mention the issue of parameterizing these models (either simple or complex) with real single-scale and multi-scale data, which is mainly due to the lack of such data.The author confirms being the sole contributor of this work and has approved it for publication.RE acknowledges support from a French Agence Nationale de Recherche (ANR) grant number ANR-21-CE45-0025-01.The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Editorial: Insights in mathematical biology 2022

The eukaryotic cell cycle comprises 4 phases (G1, S, G2, and M) and is an essential component of cellular health, allowing the cell to repair damaged DNA prior to division. Facilitating this processes, p53 is activated by DNA-damage and arrests the cell cycle to allow for the repair of this damage, while mutations in the p53 gene frequently occur in cancer. As such, this process occurs on the cell-scale but affects the organism on the cell population-scale. Here, we present two models of cell cycle progression: The first of these is concerned with the cell-scale process of cell cycle progression and the temporal biochemical processes, driven by cyclins and underlying progression from one phase to the next. The second of these models concerns the cell population-scale process of cell-cycle progression and its arrest under the influence of DNA-damage and p53-activation. Both systems take advantage of structural modeling conventions to develop novels methods for describing and exploring cell-cycle dynamics on these two divergent scales. The cell-scale model represents the accumulations of cyclins across an internal cell space and demonstrates that such a formalism gives rise to a biological clock system, with definite periodicity. The cell population-scale model allows for the exploration of interactions between various regulating proteins and the DNA-damage state of the system and quantitatively demonstrates the structural dynamics which allow p53 to regulate the G2- to M-phase transition and to prevent the mitosis of genetically damaged cells. A divergent periodicity and clear distribution of transition times is observed, as compared with the single-cell system. Comparison to a system with a reduced genetic repair rate shows a greater delay in cell cycle progression and an increased accumulation of cell in the G2-phase, as a result of the p53 biochemical feedback mechanism.

The eukaryotic cell cycle comprises 4 phases (G1, S, G2, and M) and is an essential component of cellular health, allowing the cell to repair damaged DNA prior to division. Facilitating this processes, p53 is activated by DNA-damage and arrests the cell cycle to allow for the repair of this damage, while mutations in the p53 gene frequently occur in cancer. As such, this process occurs on the cell-scale but affects the organism on the cell population-scale. Here, we present two models of cell cycle progression: The first of these is concerned with the cell-scale process of cell cycle progression and the temporal biochemical processes, driven by cyclins and underlying progression from one phase to the next. The second of these models concerns the cell population-scale process of cell-cycle progression and its arrest under the influence of DNA-damage and p53-activation. Both systems take advantage of structural modeling conventions to develop novels methods for describing and exploring cell-cycle dynamics on these two divergent scales. The cell-scale model represents the accumulations of cyclins across an internal cell space and demonstrates that such a formalism gives rise to a biological clock system, with definite periodicity. The cell population-scale model allows for the exploration of interactions between various regulating proteins and the DNA-damage state of the system and quantitatively demonstrates the structural dynamics which allow p53 to regulate the G2- to M-phase transition and to prevent the mitosis of genetically damaged cells. A divergent periodicity and clear distribution of transition times is observed, as compared with the single-cell system. Comparison to a system with a reduced genetic repair rate shows a greater delay in cell cycle progression and an increased accumulation of cell in the G2-phase, as a result of the p53 biochemical feedback mechanism.

Structured dynamics of the cell-cycle at multiple scales

The drive to maximize food production in a sustainable manner is a paramount concern for farmers and governments. The aim of food producers is to maximize their production yield employing actions such as application of fertilizer or pesticide they believe help to achieve this aim. However, farms do not exist in isolation, but rather share a landscape with neighbors forming networks where any action taken by any one farmer affects their neighbors who are forced to take mitigating actions creating a complicated set of interactions. Understanding these [non-]cooperative interactions and their effect on the shared ecosystem is important to develop food security strategies while protecting the environment and allowing farmers to make a living. We introduce a simple competitive agent based model in which agents produce food that is sold at a fixed price (we ignore market dynamics and do not include explicit punishment on any agent). We analyzed agent's profits in several simple scenarios allowing us to identify the most advantageous set of actions for maximizing the yield (and thus profit) for each farmer. We show that the effect of the structure of the network on each farm has implications on the actions taken by agents. These results have implications for the understanding of the effects of farming practices on the environment and how different levels of cooperation between farmers, taking into account the local terrain, can be used to incentivise producers to minimise the effects on the environment while maximizing yields.

The drive to maximize food production in a sustainable manner is a paramount concern for farmers and governments. The aim of food producers is to maximize their production yield employing actions such as application of fertilizer or pesticide they believe help to achieve this aim. However, farms do not exist in isolation, but rather share a landscape with neighbors forming networks where any action taken by any one farmer affects their neighbors who are forced to take mitigating actions creating a complicated set of interactions. Understanding these [non-]cooperative interactions and their effect on the shared ecosystem is important to develop food security strategies while protecting the environment and allowing farmers to make a living. We introduce a simple competitive agent based model in which agents produce food that is sold at a fixed price (we ignore market dynamics and do not include explicit punishment on any agent). We analyzed agent&apos;s profits in several simple scenarios allowing us to identify the most advantageous set of actions for maximizing the yield (and thus profit) for each farmer. We show that the effect of the structure of the network on each farm has implications on the actions taken by agents. These results have implications for the understanding of the effects of farming practices on the environment and how different levels of cooperation between farmers, taking into account the local terrain, can be used to incentivise producers to minimise the effects on the environment while maximizing yields.

A toy model of food production in a connected landscape

Severe burn injuries often lead to skin contraction, leading to stresses in and around the damaged skin region. If this contraction leads to impaired joint mobility, one speaks of contracture. To optimize treatment, a mathematical model, that is based on finite element methods, is developed. Since the finite element-based simulation of skin contraction can be expensive from a computational point of view, we use machine learning to replace these simulations such that we have a cheap alternative. The current study deals with a feed-forward neural network that we trained with 2D finite element simulations based on morphoelasticity. We focus on the evolution of the scar shape, wound area, and total strain energy, a measure of discomfort, over time. The results show average goodness of fit (R2) of 0.9979 and a tremendous speedup of 1815000X. Further, we illustrate the applicability of the neural network in an online medical app that takes the patient's age into account.

Severe burn injuries often lead to skin contraction, leading to stresses in and around the damaged skin region. If this contraction leads to impaired joint mobility, one speaks of contracture. To optimize treatment, a mathematical model, that is based on finite element methods, is developed. Since the finite element-based simulation of skin contraction can be expensive from a computational point of view, we use machine learning to replace these simulations such that we have a cheap alternative. The current study deals with a feed-forward neural network that we trained with 2D finite element simulations based on morphoelasticity. We focus on the evolution of the scar shape, wound area, and total strain energy, a measure of discomfort, over time. The results show average goodness of fit (R2) of 0.9979 and a tremendous speedup of 1815000X. Further, we illustrate the applicability of the neural network in an online medical app that takes the patient's age into account.