By Matthew R. Francis
Human beings are not bicycles. However, mechanistic metaphors for the human body abound. For instance, we compare athletes to finely-tuned machines and look for equations that are derived from mechanics to describe biological processes — even when the relationship is no better than an analogy.
However, the concept of homeostasis clearly exemplifies the breakdown of mechanistic models when one applies them to the human body. Homeostasis is the process by which an organism maintains a stable output regardless of input (within reasonable limits). The most familiar example is human body temperature, which stays within a remarkably small range of values regardless of whether one is sitting in a cold room or walking outside on a hot day.
“In a bicycle, you know what each part is for,” Michael Reed, a mathematician at Duke University, said. “We are not machines with fixed parts; we are a large pile of cooperating cells. The question is, how does this pile of cooperating cells accomplish various tasks?”
More specifically, how can researchers mathematically describe the function of these processes? Homeostasis is the conceptual opposite of bifurcation, wherein a small variation in initial conditions results in a massive change to the system’s evolution. But it is also unlike dynamical equilibrium, in which system perturbations often lead to oscillations.
All living things rely on homeostatic mechanisms, and homeostasis is an essential component of a wide range of biological and biomedical phenomena, including cancer growth, physiological responses to drugs, and hormone therapies. Mathematical modeling of these systems—in tandem with animal experiments—promises new ways to treat diseases and imbalances, both by identifying healthy homeostatic schemes and disrupting homeostasis in pathogens or tumors.
“Not only are these very complicated biological mechanisms, but overlaid on top of the biochemistry and physics are various control mechanisms that adjust for this kind of variation,” Reed said. During his presentation at the 2021 American Association for the Advancement of Science (AAAS) Annual Meeting, which took place virtually this February, Reed noted that the amount of a particular enzyme that is produced in the liver can vary as much as 25 percent without affecting liver function. Such preservation requires a fine level of control that results from a complex nonlinear relationship between input, regulation, and output functionality.
“Homeostasis is a biological concept, [so] you have to make it into a mathematical concept,” Martin Golubitsky, a mathematician at Ohio State University who also spoke at the AAAS session, said. “It’s a really difficult mathematical problem analytically; what does it mean to be ‘approximately constant’? You know it when you see it to some extent, but you don’t know how to search for it very easily.”
Homeostasis Giveth and Homeostasis Taketh Away
Human beings are like bicycles, at least in a dynamical sense. The forward motion that stems from pedaling helps to maintain balance and keep the bicycle from falling over. Similarly, equilibrium for a living thing is death; homeostasis upholds the balances that keep organisms alive and healthy.
Figure 1. Homeostasis of dopamine under a wide range of production of the enzyme tyrosine hydroxylase (TH) and the dopamine reuptake transporter (DAT), which are to varying degrees controlled by genes. The stated values are all relative to an average genotype, which the authors of the study call the “wild type.” Figure courtesy of [1] and reproduced with permission under the
Creative Commons Attribution 4.0 International License.
Reed’s research involves the creation of mathematical models that analyze the body’s regulation of dopamine — a chemical that helps transmit signals between cells, particularly in the nervous system. Low dopamine levels are a key factor in Parkinson’s disease, while hyperactive dopamine receptors
may be associated with schizophrenia. Reed and his collaborators found that their model displayed an impressive stability in dopamine levels across a wide range of genetic variations that produce different levels of regulatory enzymes (see Figure 1). This stability serves as a strong demonstration of homeostasis in the dopamine cycle [1].
In some cases, immune systems can hijack homeostatic processes to fight pathogens. Reinhard Laubenbacher, director of the Laboratory for Systems Medicine at the University of Florida, is particularly interested in the way in which cancer disrupts iron metabolism and the immune system’s ability to stop fungal infections. During his AAAS talk, Laubenbacher described how certain fungi parasitize iron from lung cells. Under ordinary circumstances, however, the immune system disrupts homeostasis in the invader cells’ iron metabolism to starve them of this vital element and ultimately kill them.
This mechanism for interrupting homeostasis provides researchers with possible alternative treatment options beyond antifungal medication. Like antibiotics, many antifungals are becoming less effective as fungi evolve. “We work very closely with immunologists,” Laubenbacher said. “They might say, ‘An immunocompromised patient doesn’t have a certain type of white blood cell, so how can we make up for it? What if we inject some more of this kind of substance, would that make a difference?’ In the [computer] model, we can do that and see if it does make a difference.”
Such results inform laboratory experiments that can lead to new treatments, particularly when other therapies are ineffective. While administering drugs to an already weakened patient might be harmful, interventions that restore homeostasis after its disruption could be possible.
Networks and Nodes
Human beings are like networks. Despite having individual roles, cells work together to ensure the function of organs and the entire body. The output of a single essential biochemical might involve many cell types across multiple organs, but sometimes one can abstract the process when studying homeostasis. A three-node network that consists of a single input \(\iota\), output \(o\), and regulatory node \(\rho\) serves as a simple example. Researchers can model networks of these three nodes to produce three types of homeostasis, one of which corresponds conceptually to the dopamine regulatory system that Reed and his colleagues described.
Consider a network wherein a given input \(I\) leads to output \(x_o(I)\). Homeostasis occurs when \(x^\prime_o(I_0)=0\) for some input value \(I_0\) (the derivative is with respect to \(I\)). One can write the three-node network generically as a system of three differential equations:
\[\dot{x}_\iota=f_\iota(x_\iota,x_\rho,x_o,I)\]
\[\dot{x}_\rho=f_\rho(x_\iota,x_\rho,x_o)\]
\[\dot{x}_o=f_o(x_\iota,x_\rho,x_o).\]
The choice of functions \(f_k\) describes specific biochemical networks. For instance, for a “feedforward” excitation in which the biochemical substrate activates an enzyme that removes a product,
\[\dot{x}_\iota=f_\iota(x_\iota,I)=I-g_1(x_\iota)-g_4(x_\iota)\]
\[\dot{x}_\rho=f_\rho(x_\iota,x_\rho)=g_1(x_\iota)-g_2(x_\rho)-g_5(x_\rho)\]
\[\dot{x}_o=f_o(x_\iota,x_\rho, x_o)=g_2(x_\rho)-h(x_\iota)g_3(x_o).\]
Particular choices of the functions \(\{g_n,h\}\) lead to homeostatic behavior. Other homeostatic networks involve different combinations of variables and functions [2].
Three-node networks are mathematically tractable. However, they are not descriptive of many realistic systems, which could have as many as 50 nodes. “How many four-node networks are there? 199!” Golubitsky said. “How many different homeostasis configurations are there? 20. That’s huge!”
However, some of these seemingly complicated systems might reduce through symmetries or redundancies. Golubitsky and his collaborators are investigating this possibility by drawing on graph theory and catastrophe theory for guidance.
So Very Simple, Only a Child Can Do It
Human beings are not actually networks. One cannot reduce Laubenbacher’s models for homeostasis in iron metabolism to the type of differential equations that Golubitsky and Reed use — at least not yet. Laubenbacher also distinguishes between complicated and complex systems; though complex systems may be conceptually simple, complicated homeostatic systems often require greater levels of detail.
“[Our model] is a multiscale model,” Laubenbacher said. “It has intracellular networks, tissue-level phenomena, [and] a whole-body component. It’s made up of molecule diffusion, partial differential equations, and cells that are moving around. The model is really a hybrid of mechanistic and phenomenological modeling and has altogether maybe 150 variables.”
Laubenbacher then paraphrased Ludwig Wittgenstein’s aphorism, “Whereof one cannot speak, thereof one must be silent,” and added his own interpretation. “Mathematics provides a language for you to formulate the properties of the systems that you encounter in the life sciences,” he said. “The goal is to be really translational. We would actually like to say that if you treat these patients in this particular way, it’s going to make a difference.”
Homeostasis can simultaneously be similar and dissimilar to both bicycles and networks. Researchers might need to utilize novel mathematics to grasp these contradictions, simply because life is not mechanical. As with the early years of nonlinear dynamics and chaos theory, much current work in this field involves categorizing and searching for global patterns that suggest the underlying order. In the end, the mathematics of homeostasis may lead to a new and deeper understanding — and ultimately save lives.
References
[1] Nijhout, H.F., Best, J.A., & Reed, M.C. (2015). Using mathematical models to understand metabolism, genes, and disease. BMC Biol., 13, 79.
[2] Golubitsky, M., Stewart, I., Antoneli, F., Huang, Z., & Wang, Y. (2020). Input-output networks, singularity theory, and homeostasis. In O. Junge, O. Schütze, G. Froyland, S. Ober-Blöbaum, & K. Padberg-Gehle (Eds.), Advances in dynamics, optimization, and computation. Cham, Switzerland: Springer.
Matthew R. Francis is a physicist, science writer, public speaker, educator, and frequent wearer of jaunty hats. His website is BowlerHatScience.org.