Agent-Based Dynamics Overview:

Self-organization and pattern formation occur across biological applications at a range of length scales (e.g., within cells, at the level of cells, and at the scale of organisms). Depending on the specific problem, this collective behavior may involve stochastic fluctuations in the number of agents, long-range interactions, or domain growth. I enjoy working closely with each application to develop and analyze agent-based models that combine these stochastic and deterministic features.

Pattern formation on the body of zebrafish

I am interested in understanding how independent agents self-organize to produce macroscopic patterns, particularly during the early development of organisms. My focus has been on zebrafish (Danio rerio), a black and gold fish with biomedical and evolutionary applications. Its characteristic stripes form as the fish develops due to the self-organizing interactions of several types of pigment cells. My goal is to identify wild-type interactions, help experimentalists link genetic changes to altered cell behavior in mutations, and shed light on how fish with different patterns in the Danio genus are related.

Because patterns emerge from cell interactions, an agent-based approach is natural: we model cells as point masses and couple deterministic movement (by ODEs) with noisy rules for cell birth, death, and transitions in type on growing 2-D domains. Our models allow us to propose unknown signals behind cell behavior and offer experimentally-testable predictions about Danio evolution and zebrafish mutations.

Collaborators: Björn Sandstede (Brown University)


Pattern formation on zebrafish fins

While wild-type zebrafish feature stripes across both their body and fins, some mutations alter body patterns without impacting fin stripes. Moreover, only two types of cells (black melanophores and gold xanthophores) seem to contribute to patterning on the tailfin, but three types are involved on the body. Motivated by these observations, here we extend our original two-cell model to simulate the development of stripes on the tailfin. We study how different means of skin growth impact the patterns that develop due to the birth, death, and movement interactions of black and gold cells on a tailfin-shaped domain.

Madeline Abbott* (University of Michigan), Dorothy Catey* (Emsi), Neil Chandra* (Brown University), Bethany Dubois* (D.E. Shaw), Francesca Lim* (Brown University), Björn Sandstede (Brown University)

* Asterisk denotes undergraduate students

  • A Volkening, M Abbott*, D Catey*, N Chandra*, B Dubois*, F Lim*, B Sandstede.
    Reconciling stripe formation on the body and fins of zebrafish. In preparation.

Varicosity formation in neurons

Together with Chuan Xue, I am studying self-organization within nerve cells: healthy neurons typically feature smooth axons, but the presence and persistence of swollen beads (varicosities) along axons is a signature of traumatic brain injury. It has been suggested that varicosities can be viewed as intracellular traffic jams: stress may destabilize and rapidly shorten intracellular roadways (microtubules), leading to localized build-up of cargoes. To test this hypothesis, we are developing agent-based models of microtubule and cargo interactions in a 3-D axon domain. Our goal is to explore what selects where varicosities appear on the axon and help identify how frequency of mechanical stress determines if they persist or disappear.

Collaborators: Chuan Xue (Ohio State University)

Pedestrian movement in lecture halls

Pedestrian crowds exhibit a range of collective behaviors, including lane formation in corridors, stop-and-go waves at high density, and bi-directional movement at doorways. In this project, we focus specifically on pedestrian movement in large lecture hall settings. Motivated by new 600-person classrooms at U.C. Davis, we are extending social-force agent-based models to explore how lecture hall size impacts class turnover times. Is it possible for students to travel from their previous classes, enter a lecture hall, and find seats during the time between class? We plan to combine pedestrian movement (by SDEs) with stochastic, density-dependent rules for choosing seats within the lecture hall.

Joseph Benson (Macalester College), Mariya Bessonov (NYC College of Technology), Korana Burke (University of California, Davis), Simone Cassani (Worcester Polytechnic Institute), Veronica Ciocanel (MBI, Ohio State University), Daniel Cooney (Princeton University)

Drug resistance and cancer development

This project on tumor growth arose from the WhAM! Workshop for Women in Applied Math at the IMA in the fall of 2014. We developed a hybrid discrete-continuous model to explore how different mechanisms of drug resistance impact tumor behavior. Our model specifies PDEs for drug and oxygen diffusion from stationary blood vessels, and allows discrete cancer cells to reproduce in a 2-D domain under two conditions: acquired (drug-induced) and pre-existing resistance. Our results suggest that the tumor micro-environment has a strong impact on cancer dynamics when drug resistance is acquired.

Zahra Aminzare (University of Iowa), Jana L. Gevertz (The College of New Jersey), Kerri-Ann Norton (Bard College), Judith Perez-Velazquez (Helmholtz Zentrum München), Katarzyna A. Rejniak (Moffit Cancer Center)


Stability analysis of agent-based models with PDMPs

Agent-based models arise naturally in many different settings: pedestrians in a crowded room, shoaling fish, and cars on a road can all be studied as systems of moving agents. Often it is appropriate to consider a set number of agents (e.g., no off-ramp), but some systems couple movement with random fluctuations in population size. This summer REU project was motivated by zebrafish patterning, since our modeling work suggests that the time-scales for cell migration and birth/death are similar in this setting. We are interested in understanding the stability of zebrafish stripes directly from an agent-based perspective, and, more generally, exploring the long-term behavior of agent-based models that couple deterministic movement with random fluctuations in population size. Our work focused first on a toy model of patterning in 1-D and made use of piecewise-deterministic Markov processes (PDMPs) to describe agent dynamics.

Cassandra Cole* (Brown University), Philip Doldo* (Cornell University), Claire Qing Fan* (Pomona College), Veronica Ciocanel (MBI, Ohio State University), Björn Sandstede (Brown University)

* Asterisk denotes undergraduate students

Alexandria Volkening
Last updated Nov. 24, 2018