How Zebrafish Get Their Stripes... or Spots

By Alexandria Volkening and Björn Sandstede

When we think of self-organization, flocking birds, swarming locusts, schooling fish, or traffic flow might come to mind. It is remarkable that the interactions of individual agents can yield reliable and diverse group dynamics across these systems. This is particularly true for biological self-organization at smaller scales. For example, healthy organism development relies on the careful interactions of different types of cells, and tissue-level properties emerge robustly as organisms grow despite the inherently noisy environments in which cells operate. Our work focuses specifically on cellular self-organization in zebrafish skin patterns.

These small freshwater fish have light and dark stripes across their bodies and fins (see Figure 1b). Interacting pigment cells—which migrate, differentiate, change color or shape, and compete on the growing skin—create the characteristic stripes. Much is unknown about these interactions. For instance, what signals cause a cell to change shape? What genes control cell interactions? How do genetic mutations alter behavior — and consequently, larger organism-scale features? We use modeling to predict the unknown cues that underlie newly-observed cell behaviors in zebrafish.

From a mathematical perspective, zebrafish skin patterning is a rich problem because it involves cell interactions at short and long range [7], and self-organization occurs as the fish grows (see Figures 1a and 1e). Zebrafish mutants also display a large range of altered skin patterns, including spots, labyrinth curves, and broken stripes. To better understand these patterns, modelers have focused on the interactions of two cell types: black melanophores and gold xanthophores. Previous models have included agent-based [2, 12], cellular automaton [1], and continuum approaches [4, 7], and these various techniques work together to provide different perspectives on pattern formation.

Figure 1. Self-organization on zebrafish skin. 1a and 1b. Stripes emerge on growing fish skin due to interactions among three main types of pigment cells: melanophores, xanthophores, and iridophores [10]. 1c and 1d. In the absence of iridophores, spots form on the body of the shady mutant. 1e. Cell interactions may involve long extensions to mediate communication at a distance. 1f. We model cell communication by considering the types of cells that fall within a long-range annulus (red) and short-range disk (blue), which are centered at a cell of interest. Figures 1a and 1f adapted from [13], 1b and 1c courtesy of [3], 1d courtesy of Alexandria Volkening and Björn Sandstede, and 1e adapted from [5]. 1a and 1f are licensed under CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/); and 1b, 1c, and 1e are licensed under CC-BY 3.0 (http://creativecommons.org/licenses/by/3.0) and published by the Company of Biologists Ltd.

New Cellular Players in Pattern Formation

Since the development of the aforementioned models, the empirical view of zebrafish patterns has undergone a paradigm shift [6, 8]. Until about 2014, the biological community focused only on melanophores and xanthophores. Researchers acknowledged a third type of pigment cell, called the iridophore, but assumed that these cells were unrelated to pattern formation because they are spread across the fish’s entire body. Experimentalists have since observed that iridophores take on two different forms in light and dark stripes, and—strikingly—if iridophores fail to appear, zebrafish develop spots instead of stripes (see Figures 1c and 1d) [3, 9, 11].

To form stripes, iridophores diffuse outward from a central horizontal marker on the body [3, 11]. As they spread, they transform in color and shape between dense (silver) and loose (blue) (see Figure 1a). These transitions are critical: when dense iridophores become loose, they signal the formation of a new dark stripe; black melanophores then emerge in this area. On the other hand, gold xanthophores react and new light stripes emerge in places where spreading loose iridophores become dense.

It is now clear that iridophores help direct black and gold cells by changing their shape and color. But what signals instruct them to do so? To help answer this question, we developed an agent-based model [13] that accounts for all three main types of pigment cells. We included two subtypes of xanthophores and iridophores, allowing these cells to adopt dense or loose forms (see Figure 1a). Broadly, we model cells as point masses and couple deterministic migration with stochastic, discrete-time rules for cell birth, death, and shape/color changes. Our model simulates the timeline of pattern development on growing domains (see Figure 1a).

We describe cell migration with ordinary differential equations using a kinematic approximation; for example, the movement of the \(i\)th black cell with position \(\mathbf{M}_i(t)\) is given by

\[ \frac{d\mathbf{M}_i}{dt} = \underbrace{\sum_{\text{other} \: M \: \text{cells}} F^{MM}(||\mathbf{M}_j - \mathbf{M}_i||)\frac{\mathbf{M}_j -\mathbf{M}_i}{||\mathbf{M}_j-\mathbf{M}_i||}}_{\text{repulsion from melanophores} \: (M)}  + \underbrace{\sum_{X^{\text{d}}\text{ cells}} F^{X^{\text{d}}M}(||\textbf{X}^{\text{d}}_j - \textbf{M}_i||)\frac{\textbf{X}^{\text{d}}_j -\textbf{M}_i}{||\textbf{X}^{\text{d}}_j-\textbf{M}_i||}}_{\text{repulsion from dense xanthophores} \: (X^{\textrm{d}})} + \]

\[\underbrace{\sum_{I^{\text{d}}\text{cells}} F^{I^{\text{d}}{M}}(||\textbf{I}^{\text{d}}_j - \textbf{M}_i||)\frac{\textbf{I}^{\text{d}}_j -\textbf{M}_i}{||\textbf{I}^{\text{d}}_j-\textbf{M}_i||}}_{\text{weak repulsion from dense iridophores} \: (I^{\text{d}})}. \]

Here, \(\textbf{X}^\textrm{d}_j(t)\) and \(\textbf{I}^\textrm{d}_j(t)\) are the positions of the \(j\)th dense xanthophore and iridophore respectively, and

\[F^{\mu \nu}(d) = -\: R^{\mu \nu}\left(\frac{1}{2} + \frac{1}{2} \tanh \left(\frac{r_{\mu \nu} -d}{\delta}\right)\right) \]

is a repulsive or attractive force. Our rules for cell birth, death, and form changes are in turn given by inequalities that must be satisfied for specific behaviors to occur. These inequalities depend on the types of cells in different interaction neighborhoods. For example, we specify that the \(i\)th silver iridophore at position \(\textbf{I}^{\text{d}}_i\) becomes loose when there are sufficient black cells in a ball \((B^{\textbf{I}^\textrm{d}_i})\) centered around it:

\[ \sum_{M \: \text{cells}}{\unicode{x1D7D9}}_{{B}^{\textbf{I}^\textrm{d}_i}} (\textbf{M}_j)  > 3 \text{ cells} \: \Rightarrow \text{dense iridophore at position} \: \textbf{I}^{\text{d}}_i \: \text{transforms to loose}. \tag1 \]

Our full model [13] combines rules with this general flavor, and in some cases involves nonlinear combinations of conditions that must be met simultaneously for given cell behaviors to occur.

Specifying parameters and rules in agent-based models is often tricky. To address this challenge, we first utilized the wealth of literature on melanophores and xanthophores, the two original cell types. We based our parameters for migration, birth, and death on empirical measurements and past experiments; for these interactions, one can view our model as descriptive rather than predictive. This left us to specify the rules for iridophore-form transitions, which are poorly understood but critical for the formation of new stripes.

Puzzling out Signals that Cells Receive to Change Their Clothes

To address the paucity of information on iridophores, we split the biological data into distinct sets for model derivation and evaluation. Zebrafish display a few major types of mutant patterns, which provided a natural means of dividing the data. Studying the development of wild-type fish and mutants lacking cell types (our derivation set) allowed us to identify possible mechanisms that might be governing iridophore-form transitions. The first class of mutants features altered patterns simply because a cell type fails to appear. Only one of our proposed mechanisms was capable of reproducing our full derivation set. We checked our model by testing its performance on our evaluation set: a series of well-understood mutants and experiments. We found good agreement across the last 15 years of biological data.

Figure 2. Example of pattern diversity. Model simulations of the following: 2a. Danio margaritatus. 2b. Mutant nacre zebrafish. 2c. Wild-type zebrafish (Danio rerio). 2d. Mutant pfeffer zebrafish. 2e. Danio albolineatus. Figure adapted from [13]; individual images licensed under CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/).

We concluded that iridophores depend on a complex, nonlinear combination of redundant cues from black melanophores and gold xanthophores (at short and long range) to change their shape. From a mathematical viewpoint we were hoping for more elegant rules, so we tried breaking down our iridophore mechanisms into their simpler components, as in \((1)\). This led to an unfamiliar pattern: light spots on a dark background (see Figure 2a). Many zebrafish mutants have black spots, but none display light spots. Instead, the pattern we found resembles Danio margaritatus, a relative of zebrafish. We therefore suggest that the complex, redundant nature of iridophore interactions might provide a source of pattern variability in closely-related fish.

Striped and Spotted Fish?

Although mutant zebrafish that lack iridophores feature spots on their bodies, this does not occur on the fins: light and dark stripes form just fine on fins based on interactions between the original two cell types (see Figures 1c and 1d). We are currently exploring how different environments on the fish body and fins may lead to spots in one region but stripes in another. How do the pigment cells in spots and stripes interact across pattern interfaces? Many questions remain, and we expect that biological and mathematical perspectives on zebrafish patterning will continue to evolve together. 

References
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	Alexandria Volkening is an NSF-Simons Fellow in the NSF-Simons Center for Quantitative Biology and the Department of Engineering Sciences and Applied Mathematics at Northwestern University.
	Björn Sandstede is a professor of applied mathematics, a Royce Family Professor of Teaching Excellence, and director of the Data Science Initiative at Brown University.