A (Very Short) Essay About (Very Large) Numbers in Embryonic Development

Cameron Schmidt
Can quantitative tools such as agent-based modeling help us understand the emergence of complex functions in embryonic development?

“Only in the cooperation of large numbers do statistical laws begin to operate the behavior of ensembles. In this way, events acquire truly orderly features”. -Erwin Schrodinger, ‘What is Life?’ 1944 Starlings are small passerine birds that occupy a wide range of habitats throughout the world. They are extremely gregarious, and roost in densely packed groups. When these groups take flight, they form a massive swarm that can number in the hundreds of thousands, known as a murmuration. The birds’ collective motion forms patterns that twist, swirl, and dance- yet rarely break continuity. They are truly a sight to behold, appearing as black masses that obscure the reality their own composition. These murmuration events have entertained the minds of mathematicians for decades, in part, because it is believed that they encode insight into the fundamental rules that govern the emergence of complex features from the simple actions of self-organizing agents. In other words, murmuring is a statistical property of large numbers of birds, and its features (as Aristotle once said) exist beside the individual parts...

In 1910, the Soviet biologist A.G. Gewurtsch, inspired by the work of Wilhelm Roux, introduced the notion of morphogenetic fields to describe embryogenesis. Gewurtsch noted that even in symmetrical masses, the number of mitotic divisions occurring in different regions simultaneously were not the same, but rather, related to each other in a manner that fit a normal (Gaussian) probability distribution. This led him to conclude that the individual cell divisions were less related to one another than they were to an unidentified ‘supracellular’ ordering factor. He continued with remarkable work in several developmental systems, and by 1944 had postulated a more complete field theory law:

“A cell creates a field around it that extends outside the cell into intercellular space... Therefore, at any point of a group of cells there exists a single field being constituted of all the individual cell fields... Hence, the properties of this aggregate field will depend, besides other factors, also on the configuration of the multicellular whole.”

Gewurtsch diverged from his more ‘descriptively minded’ colleagues in that his main thrust was to evaluate quantitatively the behavior of the embryo as a complete static entity, as well as its resulting influence on its component cells. In his words:

“A process may become accessible to explanation only insofar as one can succeed in substituting a purely phenomenological multiplicity for the understanding of a less arbitrarily created picture correctly reflecting reality”. The entire process should be accessible for analysis into a finite number of stages, each stage being represented as a monotonic function of some definite initial conditions and a single variable such as time, or distance, etc. If this cannot be realized, we consider a given set of events as scientifically inaccessible”.

During the decades in which he developed these ideas, Gewurtsch insisted that this invisible field property constituted a mode of inheritance. That position set him on a bitter collision course with Thomas Hunt Morgan and his chromosomal theory of inheritance, which famously emerged around the same time. The rest, as they say, is history. The modern synthesis followed swiftly on the heels of Morgan’s discoveries. In the decades that followed, those ideas percolated into a modern consensus that is defined not only by the idea that genes are fundamental units of inheritance, but also that individual cells (as a result of their genes) serve to direct the collective process of embryogenesis through complex morphogen-dependent signaling circuits…

During the mid-20th century, researchers investigated starling behavior as a social construct. Noting that densely packed roosting sites would trend toward an evenly spaced pattern (~4” between each bird). This structure was highly dynamic and tended toward an equilibrium with small perturbations resulting in some birds shuffling around to redistribute and others leaving and returning to the roost in a new position. In other words, the patterning was largely determined by the individual interests of the birds, motivated by antagonistic social forces (i.e. being close, while simultaneously distanced). Murmuring behavior, however, was more difficult to measure because of its dynamism.

A breakthrough came in 1987 when Craig Reynolds published an agent-based computer model known as the ‘Boids’ model. The model reproduced the apparent complexity of a murmuration by scaling just three simple behaviors among each participating bird: 1) separation (avoid crowding), 2) Alignment (navigate toward average neighbor heading), and 3) Cohesion (steer toward average position of neighbors). Thus, each bird controls the unitary behavior of the flock and is in turn controlled by it.

The Boids model was useful in reproducing simulated behavior that looked like murmuration, and its basic tenets were largely accepted following the advent of high-speed cameras, which facilitated experimental testing. The key interpretation of the model is that the birds participate as individuals, but their immediate behaviors are determined by a fixed radius of neighbors in their vicinity. Astoundingly, the ability of the flock to respond to a perturbation (e.g. a predator attack), is faster than if the information was transmitted from neighbor-to-neighbor in a simple network fashion. This property mathematically converges. And so, as the size of the flock increases, so does the reaction speed…

Embryogenesis and starling murmuration are clearly not the same thing. For starters, murmuration produces chaotic structure, while embryogenesis produces structure that (ideally) varies minimally. However, they do share one very important property- they both arise without any apparent centralized control. Today, exciting new technologies including directed gene editing, cellular barcoding, single cell ‘-omics’, and organoid subculture are advancing our ability to understand how individual cell traits are distributed in complex tissues. But these investigations will most likely be guided by the prevailing view that tissue patterning/behavior can be reduced to the regulation and function of individual (or collections of) genes. The role of the ‘environment’ may be implied, but there are far fewer models that quantitate the environmental role in development than those that describe the dynamics of intracellular morphogen signaling. Perhaps there is more to be learned by seeing developmental processes through the lens that Gewurtsch viewed them, as a quantitative property that exists somewhere between the individual and the collective...

Additional Resources:
Agent-based models are an excellent ‘low threshold’ ‘high ceiling’ way to incorporate mathematical modeling into your research. Agent-based models attempt to explain complex empirical observations through the implementation of simple rules. They are also a fantastic teaching tool. If you would like to learn more about agent-based modeling, there is a recent review by Glen et al., 2019 to check out. Additionally, I recommend downloading the free programming environment Netlogo- https://ccl.northwestern.edu/netlogo/. After you download, check out file > models library. Within each model there is a documentation tab that provides a complete description of the model and how it works. The Boids model described in this essay, is called the ‘flocking’ model in Netlogo. If you would like some guided learning in agent-based modeling, check out the Santa Fe Institute’s Complexity Explorer website- https://www.complexityexplorer.org/. There you will find tons of free online course content that covers important topics in complexity science. In the archived courses, there is an excellent agent-based modeling course taught by Bill Rand at NC State University- https://www.complexityexplorer.org/courses/121-introduction-to-agent-based-modeling.

1. Beloussov, Lev V., John M. Opitz, and Scott F. Gilbert. "Life of Alexander G. Gurwitsch and his relevant contribution to the theory of morphogenetic fields." International Journal of Developmental Biology 41.6 (2004): 771-777.
2. Emlen, John T. "Flocking behavior in birds." The Auk 69.2 (1952): 160-170.
3. Chazelle, Bernard. "The convergence of bird flocking." Journal of the ACM (JACM) 61.4 (2014): 1-35.
4. Glen, Chad M., Melissa L. Kemp, and Eberhard O. Voit. "Agent-based modeling of morphogenetic systems: Advantages and challenges." PLoS computational biology 15.3 (2019): e1006577.

About the author:
Dr. Cameron Schmidt is a NIH NRSA Postdoctoral Fellow at the East Carolina Diabetes and Obesity Institute at East Carolina University. His work in cellular bioenergetics and redox systems has explored metabolism-microenvironment interactions in cardiovascular disease, cancer, and metabolic disease. He is currently building an independent research program that will quantify and model these interactions during fertilization and early embryonic development. For continued access to his stream of consciousness, follow him on Twitter @CAS_mitolab and Instagram @cas_mitolab.