Motivating Dynamical Systems of Cell Fates through Waddington’s Landscape

Think about the roughly 30 trillion cells that make up your body. Each one has a certain function and plays a role in the biological systems that define you as a living organism. As a result, it only makes sense that they come in a variety of shapes and structures. The epidermal cells found in your skin are different from the myocytes that make up your muscles; these are vastly different from the lymphocytes circulating through your blood, and those are quite a bit different from the neurons in your brain. But how does such a diverse population of cells arise?

It has been proven that every nucleated cell carries its complete genome — in humans, this means each cell has all 3.8 billion DNA base pairs across 23 pairs of chromosomes. However, the reason to why each cell can have nonidentical structures and functionalities comes from each cell’s varied gene expression levels; while each cell has the full genome, only a portion of that genome will actively be transcribed into mRNA then translated into proteins for the given cell.

All cell lineages can be traced back to stem cells, which have the potential to develop into many different types of cells in the body. During development, embryos are comprised of embryonic stem cells which are pluripotent, meaning they can differentiate into any cell type in the body; as adults, we still have adult stem cells in low numbers, which are slightly more restricted in differentiation but still have high plasticity to become a multitude of cell types. A stem cell’s gene expression will determine what cell lineage it will follow, and eventually what terminal cell fate it will resolve. What determines a given cell’s respective gene expression levels are the complex systems of gene regulatory networks and feedback loops that manage the chemical signals/cytokines it receives from its neighboring cells and extrinsic cellular environment.

This begs the question: if we knew the current gene expression of a given cell, could we model its differentiation as a dynamical system of the possible phenotypical state space, and thus predict the most likely cell fates?

Waddington’s Landscape

All the way back in the 1940s and ’50s, biologist Conrad Waddington first introduced the idea of cell fate being depicted as a ball rolling down a rugged terrain, headed for valleys that represent terminal cell fates [1]. This model is coined Waddington’s landscape.

Image of Waddington’s landscape [1]

From the above image, you can imagine the ball at the very top position of the hill representing a stem cell with high differentiation potential; as it rolls down the hill, its loses plasticity as its gene expression evolves, until it finally reaches a terminal cell fate. Despite the lack of sequencing technology and contemporary understanding of the genome, Waddington laid the foundation of the theory of cell fate, and many of Waddington’s ideas are still widely accepted today.

Viewing Cell Fates in the Modern Day

Since the time of Waddington and his landscape, significant progress in the theory of dynamical systems has emerged, and we now have a better framework to quantify the dynamics of cell fate. Let’s start by going over the underlying intuition.

Let’s say we can map the gene expression of every cell to a point in a two-dimensional plane. While this is an oversimplification of the matter, you can imagine dimensionality reduction techniques, such as PCA or UMAP, as methods for plotting gene expression on a 2-D surface.

Additionally, for each point on the gene expression surface, we have an associated plasticity value, which denotes how able a cell at that gene expression is to differentiate into another phenotype. A point of high plasticity would represent a stem cell, whereas a point of low plasticity could relate to a terminal cell phenotype, or one that will not differentiate any further. In mathematical terms, if gene expression lies on the xy-plane and plasticity is represented as the function P, then plasticity can be written as a function of gene expression: P = f(x, y).

Here, we would expect a stem cell to traverse down the gradient of P until it reaches a local minima (terminal cell fate). We can also imagine that as a cell’s environment and stimuli begin to change, then the topology of P will shift accordingly.

In a more rigorous mathematical sense, we can set up a dynamical system to describe cell states in a more complete manner. A given cell state is given by an n-dimensional vector, denoted x, that represents the levels of the n genes (or any quantitative variables) that are relevant for describing a cell state. The dynamics of how each variable changes with respect to both time and x can be modeled by a set of coupled differential equations, given by F(x) [2].

Furthermore, the subspaces of the gene expression space where cells tend towards most often can be viewed as attractors in the context of dynamical systems. There can be several types of attractors, each with different mathematical properties; some of these include more predictable attractors such as fixed-point attractors and spiral sink attractors, and others being more chaotic examples such as strange attractors [2].

Following the language of dynamical systems, the system can be assumed to be chaotic, meaning slight perturbations (i.e. differences in initial conditions) may lead to large variations in downstream outcomes. For example, a small difference in the initial gene profile of a stem cell could result in its trajectory tending to towards entirely different attractor states (cell fates). To put this into biological context, this explains why hematopoietic stem cells, despite all originating from the bone marrow, can differentiate into any type of blood cell, ranging between cell fates such as lymphocytes, phagocytes, other granulocytes, and erythrocytes.

As one can speculate, the dynamics of cell fates become increasingly more complex as we strive to include higher dimensions of genes and other interactions; thus, it is a current research challenge to find ways to distill the key patterns of cell differentiation into feasible state space sizes [3].

Bifurcations and Control

Now that we understand how we can model the dynamics of cell fate, is it possible to actuate the system in a way that reprograms a cell to a desired outcome? In dynamical systems, a bifurcation is the qualitative change in the behavior of a system, typically as a result from a control parameter(s) being altered. In theory, if we could find the right bifurcation parameters to alter the cell dynamics landscape in a way that eliminates some cell fates or opens up others, then we could program the outcomes of cell differentiation, and thereby exploit the functionalities/characteristics of some phenotypical states over others [2].

For example, in the realm of cancer biology, tumors tend to have very heterogeneous cell populations, and are thus very difficult to treat as some subpopulations may be resistant to some therapies, whilst other subpopulations have contrasting immunities. Moreover, populations of cancer stem cells (CSCs) exists, which are malignant cells with versatile differentiation potentials. Thus, even if one treatment is able to destroy even 99% of the tumor cells in a patient, any surviving CSCs will have the potential to differentiate to other, resistant lineages. In the context of controlling cell fate, there is hope they physicians could push these CSCs with proper control mechanisms to either susceptible subpopulations, or perhaps even non-malignant cell phenotypes [4]. Therefore, continuing this line of research is critical to the future of battling cancer and other diseases.

Conclusions

The process of cell differentiation is a complex system and is dependent on many factors non-autonomous to the cell. However, following the intuition of Waddington’s landscape and the theory of dynamical systems, we are on a path to developing mathematical and computational models that will allow researchers to both predict cell fates and control the system’s dynamics. While this is a major area of on-going research, especially with the recent influx of single-cell sequencing technologies, I hope this brief introduction to the topic has provided you with a new perspective on how we can view cell differentiation.

Thank you for reading!

References

[1] Waddington, C. H. (1957). The Strategy of the Genes.

[2] Casey, M. J., Stumpf, P. S., & MacArthur, B. D. (2020). Theory of cell fate. Wiley Interdisciplinary Reviews: Systems Biology and Medicine, 12(2), e1471.

[3] Sáez, M., Briscoe, J., & Rand, D. A. (2022). Dynamical landscapes of cell fate decisions. Interface focus, 12(4), 20220002.

[4] Uthamacumaran, A. (2021). A review of dynamical systems approaches for the detection of chaotic attractors in cancer networks. Patterns, 2(4).