SIAM News Blog

Tracing Genealogy Within an Invasion Wave

By Kerry Landman

Invasion waves arise in many systems, including wound healing, brain tumor expansion, and the displacement of indigenous species by an introduced species. 

Mathematical models of invasion systems are often described by Fisher’s equation, which contains two essential mechanisms – the ability of the entities to move and increase in number through proliferation (cell division or reproduction). Fisher’s equation contains a diffusion term and a logistic growth term, which includes crowding effects involving a carrying capacity:

\[\frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2}+ \lambda C \left(1-\frac{C}{K}\right).\qquad (1) \]

Phase plane arguments demonstrate that the partial differential equation (PDE) exhibits a travelling wave solution with a minimum wavespeed \(2\sqrt{D\lambda}.\) Entities behind the wavefront are at carrying capacity, so proliferation only occurs in the region of the wavefront, causing the travelling wave to steadily advance. We call this frontal expansion. Such models are used to study population-level properties of invasion.  

Large Individual Variability Within the Predictable Travelling Wave

Figure 1. Clonal inequality as evidenced in three realizations (horizontal boundaries have periodic boundary conditions to emulate a cylindrical shell). Agents are labeled with different colors, and color is inherited by progeny. Top: moderate-size blue clone. Middle: a single blue agent has the large clone, responsible for invasion wave. Bottom: insignificant clone sizes. Image credit: B.J. Binder.
The enteric nervous system (ENS) is a large complex system—comparable in size to the spinal cord—in the wall of the gastrointestinal tract. It is responsible for normal gut function and peristaltic contraction, which forces food along the gastrointestinal tract. During vertebrate embryonic ENS development, a small population of immigrant neural crest cells (NCC) enters the stomach and progressively invades and colonizes the small and large intestine as a travelling wave over many days (mouse, chick), and weeks (human). The cells actively move in a two-dimensional cylindrical shell within the gut wall and undergo many rounds of cell division (without cell death). Our group at the University of Melbourne has a decade-long collaboration with Don Newgreen’s Embryology Laboratory at the Murdoch Children’s Research Institute on this ENS cell invasion process.

The evolution of the NCC population density is well-described by Fisher’s equation. In addition, agent-based models facilitate the probing of more detailed information on cell-cell interaction. We use a square lattice agent-based model for our NCC invasion system, where an agent represents a single NCC. One agent at most can occupy a lattice site at each timestep, defining an exclusion process. We then assign probabilities to local rules describing cell motility and proliferation. Using these simple simulation rules, a single realization produces a right-moving wave (with seeding agents on the left). Averaging over many simulations leads to a predictable travelling wave, analogous to Fisher’s wave.

We label agents with different colors to observe the interactions between various parts of the invasion wave. Progeny inherit the color of the parent agent, and all agents follow the same local rules. Every realization is slightly different, but at the population level each demonstrates frontal expansion as a result of progeny from the red and blue agents (see Figure 1).

In these models, we label every starting agent and determine the genealogy. The clonal contribution of each agent is highly variable, from minimal contribution to a few clones of overwhelming size, which we term superstars. We set out to determine how common these behaviors are, and if they could be shown experimentally. 

Superstars are apparent in every realization (see Figure 2). This behavior is the result of stochastic competition for space. All agents have the same ability to move and divide, but most become blocked by surrounding agents and thus can no longer do so. There is nothing inherently different about superstars. They just got lucky. 

Figure 2. Invasion wave and spatial distribution of agent tracings. (a) depicts the initial condition with 500 agents. (b) and (c) show two realizations of the travelling wave that moves progressively to the right, illustrating the largest and second largest single agent lineage tracing (pink and turquoise respec- tively) and the 498 other agent lineage tracings (all collected together in blue). In (b) there are significant differences in the agent numbers between the two largest tracings, while in (c) the two largest tracings have a similar number of agents. Image credit: B.L. Cheeseman.
Using ‘cloning-in-a-crowd’ experiments in gut explants, Dr. Newgreen’s lab showed that in a crowd of unlabeled NCC, a single cell lineage can be traced using a green marker inherited in cell division. After several days, the whole gut is full of NCC. While most experiments had very few green-labeled cells, one showed a sea of green cells. This is the biological representative of a superstar – it is the cell that yields a disproportionately large contribution to the final population. These experiments show that clonal inequality is a reality, and that superstars do exist.

It is impossible to image all cells in the invading gut system, even with multicolor techniques. Hence, we can only trace a single lineage and collect data at a single time point in the gut experiments, unlike our agent-based models. However, researchers are currently developing techniques that could provide information on the number of cells in each generation.

In our agent-based models, we can color code the agents in the same generation, instead of visualizing individual lineages. Starting with generation-zero agents (as in Figure 2a), even a single realization shows that organization occurs within the spatial distribution of generation number during invasion, despite the large spatial variability of individual lineages. Can this organization be described with PDEs?

PDEs Describing Generation Number Density

We developed a new system of PDEs to describe each cell generation number [2], which we derived from probability arguments and mean-field approximations. We considered how the average occupancy of each generation at a lattice site changes over a single timestep, while accounting for agents that leave and enter the site and noting that agents can only move into unoccupied sites. The mean-field approximation assumes that the occupancy status of neighboring sites is independent. Taking Taylor series expansions and the continuum limit leads to a coupled system of PDEs describing the generation number density \(n_i(x,t)\) in terms of the total density \(C\):

\[ \frac{\partial n_i}{\partial t} = D\frac{\partial }{\partial x}\left((1-C)\frac{\partial n_i}{\partial x}+n_i\frac{\partial C}{\partial x}\right) +\lambda (2n_{i-1}-n_{i}) (1-C),\]

\[\frac{\partial n_0}{\partial t} = D\frac{\partial }{\partial x}\left((1-C)\frac{\partial n_0}{\partial x}+n_0\frac{\partial C}{\partial x}\right) -\lambda n_0 (1-C). \]

Terms with \((1-C)\) arise due to the exclusion process, and the convective term arises due to the multi-species nature of the system. Of course, summing over all the generations gives Fisher’s equation (1). Solving this numerically, we obtain the spatial distribution of generation number density; it increases in an organized manner from left to right (see Figure 3).

Can the PDEs Predict Superstars?

Figure 3. Spatial distribution of PDE solutions for the generations \(n_i(x, t) (i = 0, 1, 2 . . .)\) The generation number \(i\) increases from left to right, each marked with a different color. The earlier generations, which correspond to those well behind the wavefront, reach a steady state. The later generations, at the wavefront, continue to evolve for some further time. Image credit: B.L. Cheeseman.
From the PDE solutions, is it possible to predict lineage variability and the existence of superstars? To do so, we consider the flow of cells from one generation to the next at time \(t.\) From the spatial distributions, we know the number of  cells in each generation at time \(t.\) Mass balance arguments determine a relationship between the number of cells and the number of cells that have undergone division in each generation. An explicit formula for the generation transition probabilities is obtained in terms of the number of cells in each generation, which allows us to define a Galton-Watson process, appropriate for cell division [2], and subsequently generate cell lineage data.

For each branching process, the simulation runs until all the trees terminate. We use a Lorenz curve [3], which commonly measures inequality of wealth distribution in econometrics. For example, a plot of the cumulative proportion of wealth versus the cumulative proportion of the U.S. population gives a curve that is far from a 45-degree straight line. In our cellular context, we look at the number of initial cells and ask how much their Galton-Watson-generated lineages contribute to the total final number of cells. It is highly unequal, and it grows more unequal as the cell proliferation rate increases. The progeny of a few superstars dominate the final population.

This method, using solutions to PDEs, has provided a genealogy with highly asymmetric lineages. It correctly identifies the existence of superstars and associated properties. These results compare very well with agent-based lineages. 

Within an embryo, the gut tissue is growing everywhere during the development of the ENS. Adding domain growth to our PDE models generalizes all the methods nicely. Furthermore, this technique for determining individual data works for other PDEs that describe motility and proliferation events. 

There is much interest in clonal advantage through mutation in the cancer field, from the viewpoint of ‘expansion of the fittest.’ We have demonstrated differential clonal expansion in an invading cell population through PDEs, agent-based models, and experiments, and argue that luck (expansion of the luckiest) may have a surprisingly large effect on differential clonal expansion.


[1]  Cheeseman, B.L., Zhang, D., Binder, B.J., Newgreen, D.F., & Landman, K.A. (2014). Cell lineage tracing in the developing enteric nervous system: superstars revealed by experiment and simulation. J. R. Soc. Interface, 11(93), 20130815. 

[2]  Cheeseman, B.L., Newgreen, D.F., & Landman, K.A. (2014). Spatial and temporal dynamics of cell generations within an invasion wave: a link to cell lineage tracing. J. Theor. Biol., 363, 344-356. 

[3] Gastwirth, J.L. (1971). A General Definition of the Lorenz Curve. Econometrica, 39, 1037-1039.

Kerry Landman is a professor in the School of Mathematics and Statistics at the University of Melbourne, Australia. In 2014 she received the Australian and New Zealand Industrial and Applied Mathematics Society’s ANZIAM medal in recognition of her contributions to industrial and applied mathematics. 


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