By Matthew R. Francis
A white blood cell slips through the gaps between other cells, stretching and bending as it goes. Though its movement strongly evokes that of a macroscopic creature—perhaps a rodent nosing its way through a maze—the cell is guided only by chemical signals and molecular forces. It has no need for a brain, not even the one in the human body that it shares.
Mathematical biologists have developed a number of models to understand self-organization both within and between cells. Leah Edelstein-Keshet of the University of British Columbia received SIAM’s prestigious 2022 John von Neumann Prize for her significant contributions to this field. Edelstein-Keshet has been a leader in mathematical biology research for several decades and also penned one of the earliest textbooks on the subject: Mathematical Models in Biology [1]. She delivered the associated prize lecture at the hybrid 2022 SIAM Annual Meeting (AN22), which took place in Pittsburgh, Pa., this July.
“I started off by looking at the interesting patterns that cells make,” Edelstein-Keshet said. “Fibroblasts try to align in parallel patterns, and the question was, how do they form these parallel arrays? We developed some mathematical language to deal with that. And it turns out that there are a lot of related problems of units that line up in parallel arrays.”
Specifically, Edelstein-Keshet and her collaborators applied similar mathematical models to actin fibers within individual cells. Actin gives cells their structure and—in concert with the molecule myosin—helps control how they move and change shape. In other words, the key to cell motion and organization lies partly in the self-organization of such fibers.
Many cells do not have a distinct front or back when in a quiescent (inactive) state, but sometimes cells polarize and reorganize their actin fibers. As in the aforementioned white blood cell example, the “front” of the cell can switch places with the back or sides to let the cell find a path; therefore, the organization of actin for steering purposes depends on stimuli both inside and outside of the cell.
Figure 1. A bifurcation diagram for the one-dimensional cell model that shows the effects of varying total protein concentration \(K\) and parameter \(\epsilon\), the latter of which depends on cell size and diffusion rate of the active protein. Active protein polarization turns “on” or “off” depending on these factors. Figure courtesy of Alexandra Jilkine and Leah Edelstein-Keshet.
During her talk, Edelstein-Keshet recognized the work of forces that are similar to those that govern pattern formation on larger scales — a concept that was first described mathematically by Alan Turing. “But it has a certain twist to it,” she said. “These proteins can interact with each other. And they can diffuse both
along the membrane of the cell and
inside the fluid part of the cell. It turned out [to be] a very intriguing new phenomenon.”
Edelstein-Keshet and her collaborators called this phenomenon “wave-pinning,” as the one-dimensional (1D) mathematical solutions describe a wave that is confined to a single part of the cell, rather than one that propagates across the entire cell. The model offers a simple explanation of the way in which certain influences could cause high molecular activity at one end of the cell—leading to movement in that direction, for example—and little to no activity at the other end.
To Be a Cell in Motion, All You Need is Some PDEs
The reaction-diffusion equations, which might look familiar from other applications in biochemistry, form the basis for the 1D wave-pinning model [2]. To describe polarization in a single protein type, the concentrations of active \(u\) and inactive \(v\) proteins are each assigned their own reaction-diffusion equation
\[\epsilon \frac{\partial u}{\partial t}=\epsilon^2\frac{\partial^2u}{\partial x^2}+f(u,v)\]
\[\epsilon \frac{\partial v}{\partial t}=D\frac{\partial^2v}{\partial x^2}-f(u,v)\]
in dimensionless form, where
\[\epsilon^2=\frac{D_u}{\eta L^2}, \enspace D=\frac{D_v}{\eta L^2}.\]
Here, \(D_u\) and \(D_v\) are the respective rates of diffusion for the active and inactive proteins, \(\eta\) is the reaction rate, and \(L\) is the cell diameter. The nonlinear coupling term \(f(u,v)\) is the difference between the activation and inactivation rates
\[f(u,v)=\left(\delta+\frac{\gamma u^2}{1+u^2} \right) v-u,\]
with positive constants \(\delta\) and \(\gamma\).
For simplicity, Edelstein-Keshet and her colleagues set Neumann boundary conditions with the cell membrane and the cytosol occupying all of the points in \(0<x<1\). The lack of flux of the protein into or out of the cell conserves the total amount \(K\):
\[\int^1_0(u+v)dx=K=\textrm{constant}.\]
Finally, the group made the reasonable assumption that the diffusion rate for the membrane-bound active protein is much lower than that of the inactive one: \(D_u\ll D_v\).
Despite its simplicity, the mathematical model also describes a mechanism—governed by parameter \(\epsilon\)—through which cells can switch polarization on and off. This parameter depends on the small diffusion rate of the active protein and is inversely proportional to the cell diameter, meaning that \(\epsilon\) itself is a small number. By examining the model’s behavior over a range of total protein concentrations and \(\epsilon\) values, the researchers generated a bifurcation diagram in which a certain range of values leads to polarization and wave-pinning. Beyond those values—i.e., at a very large diffusion rate, small cell size, or protein concentrations that are too large or small—the cell never polarizes (see Figure 1). “These are actual biological or physical properties of cells that affect whether or not they can polarize,” Edelstein-Keshet said. “And that’s in the simplest caricature of polarization.”
Figure 2. A computationally obtained bifurcation diagram for a more realistic two-dimensional system wherein the parameters are two protein activation rates. The computer model identified two bifurcation curves that separate regions where the cell exhibits spiral waves, oscillations, and other behaviors. Figure courtesy of Elisabeth Rens and Leah Edelstein-Keshet.
To move beyond that caricature, Edelstein-Keshet and her collaborators examined far more complex models. They considered a wide range of generalizations for the minimal model by allowing for the influence of other components, including multiple interacting protein species; permitting the cell to dynamically change shape; and creating two-dimensional models with and without these additional factors. Many of these generalizations were too complicated for even an indirect analytical approach and necessitated purely computational treatment [3].
“We simulated the model over and over again and extracted a summary of what we saw,” Edelstein-Keshet said. Yet clear bifurcations still emerged, even in this instance. “When you tweak certain parameters in the system, you get cells that either polarize, do nothing, exhibit some kind of oscillatory behavior, have a wave that travels through them, and so forth” (see Figure 2).
Multiscale Mathematics
These models are certainly fascinating from a mathematical standpoint, but Edelstein-Keshet also collaborates with experimental biologists to test specific predictions and assumptions. One of the biggest challenges in biological self-organization is understanding wound healing. “The details in one case were worked out experimentally,” she said. “You’ve got a cell sheet that grows and seals this open gap.”
A theoretical treatment, however, requires examining not just individual cells’ movements—for example, the means through which white blood cells find their way to the right spots—but also collective cell organizational efforts to seal the wound. Such a model must consider several scales at once, which might involve different mathematical toolkits. “We could code that into a multiscale simulation that showed under what conditions you get pieces to fragment or form fingers [to close wounds],” Edelstein-Keshet said.
Multiscale simulations are still relatively new and have only come into their own within the last decade or so. Yet despite their novelty, they have already found a range of important applications. Along with wound healing, multiscale simulations have the potential to effectively model cancers and other disorders that involve chemical signals, individual cells, tissues, and entire organs. During her AN22 lecture, Edelstein-Keshet presented state-of-the-art models that allow researchers to identify emergent behaviors from smaller-scale phenomena.
Since cellular self-organization was Edelstein-Keshet’s initial entry into this field of study, it seems fitting that she return to the topic now that computers are finally powerful enough to advance her research. “I’ve had a lot of fun working on all of these problems,” she said. “In particular, I’ve had a great deal of good fortune in having wonderful people to work with. This area has many fascinating opportunities for applied mathematicians to do things that are both interesting and hopefully scientifically useful.”
References
[1] Edelstein-Keshet, L. (2005). Mathematical models in biology. In Classics in applied mathematics. Philadelphia, PA: Society for Industrial and Applied Mathematics.
[2] Mori, Y., Jilkine, A., & Edelstein-Keshet, L. (2011). Asymptotic and bifurcation analysis of wave-pinning in a reaction-diffusion model for cell polarization. SIAM J. Applied Math., 71(4), 1401-1427.
[3] Rens, E.G., & Edelstein-Keshet, L. (2021). Cellular tango: How extracellular matrix adhesion choreographs Rac-Rho signaling and cell movement. Phys. Biol., 18(6), 066005.
Matthew R. Francis is a physicist, science writer, public speaker, educator, and frequent wearer of jaunty hats. His website is BowlerHatScience.org.