Guided by Signals: The Mathematics of Chemotaxis
by Hewan Shemtaga
Advisors: Dr. Wenxian Shen & Dr. Selim Sukhtaiev
Numerous organisms have an inherent capability to perceive external stimuli and initiate motion-based responses. For instance, microorganisms, including bacteria, exhibit the capacity to track signals to locate sustenance; immune cells intricately navigate their surroundings in response to signaling cues, directing them towards the site of infection. Similarly, sperm cells are attracted to chemical signals that get released from the outer coating of egg. Given the frequency of this phenomenon, considerable attention has been directed towards a comprehensive investigation. Advancements in the study of microscopic entities have enabled us to understand how cells and organisms read, respond, and shape sensory information derived from their environment. A significant contribution to the understanding of such intricate processes has been facilitated by the utilization of mathematical and computational modeling techniques. For years, scientific investigations have employed systems of partial differential equations (PDEs) to examine the evolving densities of cells/organisms and concentrations of chemical attractants/repellents.
Figure 1: Illustration of Chemotaxis in the presence of a Chemo-Attractant
One of the commonly recognized mathematical models of chemotaxis was introduced by Keller and Segel in 1971 [1]. This model consists of coupled partial differential equations, which characterize the evolving densities of one or more chemotactic populations along with their corresponding chemical attractants or repellents. The general form of the Keller-Segel chemotaxis model is,
Where u denotes the density of the cell/organism and v denotes the concentration of the chemical. In the above system of equations, (0.1) describes the dependence of the cell density variation over time on the migration of the population, the growth and death of the population, and the movement of the cell/organism towards or away from the chemical gradient. On the other hand, equation (0.2) represents the dependence of the chemical concentration variation over time on the diffusion of the chemical as well as the production and degradation of the chemical signal.
Our research concerns the above chemotaxis model on a one-dimensional graph (network) structure. Although the study of mathematical models on networks goes back to the 1980s, it has seen a recent surge in interest, primarily attributed to its numerous applications in the fields of biology and neurobiology. One notable application is in the domain of tissue engineering, concerning the motion of fibroblasts, the stem cells responsible for the regeneration of dermal tissue, on synthetic scaffolds during dermal wound healing. Simulating this process using a chemotaxis model on a network is suitable because the edges of the network symbolize the fibers of the scaffold and the underlying transport equations provide the evolution of fibroblast densities along each individual fiber [3]. We began by examining the validity of the general chemotaxis model on a network structure. We then proved the existence of a unique solution whose behavior changes continuously with the initial conditions. Furthermore, under realistic biological conditions, we proved solutions stay within expected biological limits at all times. Subsequently, we carried out a mathematical analysis to investigate the behavior of cells and microorganisms in response to chemotaxis where cells/organisms follow a logistic growth. Our research aimed to answer a range of biologically relevant questions, including analyzing the dynamics of the system as the chemotaxis sensitivity crosses certain thresholds. Specifically, we aimed to understand whether there are distinct shifts in the system dynamics at certain chemotaxis sensitivity thresholds, commonly referred to as bifurcation points. Our findings revealed a shift from a constant stable solution when chemotaxis sensitivity is weak, to a bifurcation state characterized by existence of other nontrivial stationary solutions in the presence of strong chemotaxis sensitivity. This outcome implies, while weak chemotaxis sensitivity does not significantly alter long-term system dynamics, the dominance of chemotaxis sensitivity over logistic growth introduces a variety of intricate and noteworthy patterns.
Figure 2: (a) and (c) show initial population of cell/bacteria on a simple network structure. (b) With weak chemotaxis sensitivity, the population stabilizes at the carrying capacity uniformly in space. In contrast, with strong chemotaxis sensitivity, initial population may converge to nonconstant stationary solution (we eventually observe pattern formations in the network structure), as depicted in (d).
In conclusion, the mathematical study of chemotaxis models not only enhances our foundational knowledge about cells and organisms behavior in response to chemotaxis but also yields practical insights applicable in fields like medicine, bioengineering, and ecology. Looking ahead, our focus will be on how chemotaxis sensitivity influences the spreading speed of cells and organisms within network structures.
References
[1] E. F. Keller, L, A, Segel, Model for chemotaxis. Journal of Theoretical Biology. 30, (1971), 225-234
[2] A. Tero, R. Kobayashi, T. Nakagaki, A mathematical model for adaptive transport network in path finding by true slime mold. J. Theoret. Biol. 244 (2007), 553-564.
[3] Bretti, G.; Natalini, R.; Ribot, M. A hyperbolic model of chemotaxis on a network: a numerical study. ESAIM: Mathematical Modelling and Numerical Analysis - Mod´elisation Math´ematique et Analyse Num´erique, Volume 48 (2014) no. 1, pp. 231-258. doi : 10.1051/m2an/2013098. http://www.numdam.org/articles/10.1051/m2an/2013098/ 3
[4] Kevin J. Painter, Mathematical models for chemotaxis and their applications in selforganisation phenomena, Journal of Theoretical Biology, Volume 481, 2019, Pages 162-182, ISSN 0022-5193, https://doi.org/10.1016/j.jtbi.2018.06.019.
[5] T. Hillen, K. Painter, A user’s guide to PDE models for chemotaxis, Journal of Mathematical Biology, volume 58 (2009), pp. 183-217. https://api.semanticscholar.org/CorpusID:249201
Acknowledgement: This material is based upon work supported by the National Science Foundation under Grant Number (NSF DMS-2243027), the Office of Vice-President for Research & Economic Development (OVPRED) through Research Support Program grant, and the Simons Collaboration Grant MP-TSM-00002897. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the funding organizations.