Native to the Himalayan region, the zebrafish (Danio rerio) is a small tropical freshwater fish belonging to the minnow family. It is popular both in recreational aquariums and as a so-called “model organism” in scientific research; scientists study the biological workings of zebrafish to gain insight into the comparable nature of other organisms. As a result, zebrafish studies have particular application to developmental biology, cancer research, and genetic disease.
Mathematical modeling of zebrafish stripes exceeds existing experimental techniques, as systematically removing or altering model components reveals novel information about the mechanisms that drive pattern formation. During a minisymposium at the 2017 SIAM Conference on Applications of Dynamical Systems, Alexandria Volkening of Brown University presented an agent-based model with a nonlocal continuum limit for zebrafish stripe mutations on a growing domain. The model explores the interplay of cells and bone rays on the body and caudal fin of the fish, with implications to the fields of biology and ecology.
Self-organizing pigment cells comprise the distinct pattern of alternating colored stripes on the body and fins of zebrafish. Image credit: Wikimedia Commons.
Self-organizing pigment cells comprise the distinct pattern of alternating colored stripes on the body and fins of a zebrafish, and are distinguishable on the skin. This lucidity facilitates Volkening’s investigation into the cells’ interaction and the resulting iconic stripe pattern. Stripes form as the fish grow and develop from larvae to adults, and begin appearing after only three weeks. “Pattern formation starts with the appearance of a light stripe at the center of the fish,” Volkening said. “As it grows, stripes are added sequentially outward.”
As with most patterns in nature, many possible mutations exist. In some cases, spots, curves, and other alternate arrays—rather than the usual stripes—appear on the body and fins of zebrafish. Volkening divided the mutations into two categories: early-stage and late-stage. In early-stage mutation, some type of pigment cell is missing. “Late-stage mutations arise when all of the pigment cells are present, but some interaction is altered,” Volkening said. “Biologists know what gene is associated with alteration, but they don’t know how that change in gene transfers into a functional difference.”
The goal of Volkening’s research is to predict the altered cell interactions in late-stage mutations. She began by establishing the biological background of pigment cells. Zebrafish have three main types of pigment cells, arranged in multiple layers on the body and fins. Black melanophores comprise the bottom later, silvery/bluish iridophores make up the middle (and give the skin its shiny appearance), and orange/yellow xanthophores are on the surface. “The cells are self-organized to create the patterns in four main ways,” Volkening said. They arrange themselves through migration, birth and death due to competition, and—in the case of the middle and surface-layer cells—form transformation.
The middle layer of cells (the silvery iridophores) is the driving force behind stripe formation. When zebrafish are three weeks old, dense white squares comprise the initial stripes. “These white squares start spreading outward,” Volkening said. “And as they spread outward, they change their form to the triangular loose form.” As growth continues, the loose triangles eventually become dense white squares again. The shift of iridophore pigments from loose to dense determines the subsequent location of melaophores and xanthophores on the body and fins.
Until fairly recently, the experimental community focused only on the migration, birth, and death of melanophores (bottom cells); they failed to acknowledge form transformation as a type of cell interaction. Consequently, very little is known about form change. To lessen this knowledge gap, Volkening put forward a predictive model to identify the factors that trigger transformation. She used an agent-based approach that examines independent pigment cells interacting on the same plane. “This allows us to work on the same time and length scales as the experimental data, and lends itself to communication with biologists,” she said. Coupled ordinary differential equations (ODEs) represent cell movement and migration, noisy discrete-time rules indicate birth and death, and discrete-time rules represent form changes.
The model’s four main goals are as follows:
- Identify wild-type cell interactions, specifically from changes of the middle and surface-level cells (iridophores and xanthophores)
- Predict altered interactions in late-stage mutations
- Suggest interactions that were lost in zebrafish relatives, and explore how patterns have evolved overtime
- Reconcile the difference in pattern formation between the body and fins
Volkening then introduced an ablation model, which uses a laser to remove an excerpt of zebrafish skin for further scrutiny. This model employs both short-term and long-term promotion and inhibition. “Deterministic rules depend on the portion of cells in short and long-range regions,” she said, and conducted training on early-stage mutations. “If we can put forth one set of rules for these set of interactions that are consistent across the fish, then we’re proving something about the biology.” This resulted in a network of short and long-range promotion and inhibition interactions. Upon establishing the network, Volkening validated her model.
When predicting late-stage mutations, Volkening investigated the appearance of spots on zebrafish bodies, a common abnormality. She spoke of a zebrafish (called Pfeffer) that lacked orange xanthophores; his development pattern looked like peppered black dots. Volkening crossed Pfeffer—a fish with an early-stage mutation—with other fish to determine which pigment properties maintain bodily patterns. While doing so, she found that some patterns are consistent with relatives of zebrafish. This realization could demonstrate how similar fish have evolved from one another.
Next, Volkening investigated the occasional discrepancies between patterns on the body and the caudal fin. She presented a zebrafish with only black (melanophore) and orange (xanthophore) cells; its body was spotted but its fins sported unmutated stripes. This inconsistency confirms that unlike the body, zebrafish fins require only two types of pigment cells for correct pattern formation. “How can we get stripes on the fins when we couldn’t get them on the body with a two-cell model?” Volkening asked. To answer this question, she developed a reduced model that included the ODEs representing migration and the noisy discrete-time rules for birth and death, but not the discrete-time rules indicating form changes.
Volkening simulated fin pattern by fitting a curve around the edge of the fin, tracing the bones, and creating a growing domain. The simulation revealed a few possible explanations of body and fin pattern disagreement. Zebrafish fins grow outward, and thus have an entirely different mechanism than the body. Additionally, if skin grows uniformly on both the body and the fins, it likely stretches more horizontally than vertically during fin growth, which might keep the pigments in a stripe formation. Furthermore, the bony fin rays presumably obstruct the movement of pigment cells up and down the fin, encouraging horizontal growth
Volkening concluded by discussing subsequent steps in the modeling process, including the exploration of mutations in other Danio fish. She also intends to more closely examine the interplay of growth and patterning on fins versus the body. Ultimately, however, her agent-based predictive model identifies relevant changes in zebrafish cellular interactions and pigmentation.
|| Lina Sorg is the associate editor of SIAM News.