Modeling parr-mark pattern formation during the early development of Amago trout

This paper studies the formation of the large dark patterns, known as parr marks, that form on the Amago trout as it grows from the early larval stages to adulthood. The Amago trout, known as Oncorhynchus masou ishikawa, exhibits stripes during the early stages of development that in turn evolve (through reorientation and peak insertion) to form zigzag spot patterns as the fish grows to adulthood. By considering a standard representation of the Turing model for biological self-organization via interacting and diffusing morphogens, we illustrate that a diffusively driven instability can generate transient patterns consistent with those experimentally observed during the process of parr-mark formation in the early development of the Amago trout. Surface evolution is modeled through an experimentally driven growth function. Our studies conclude that the surface evolution profile, the surface geometry, and the curvature are key factors that play a pivotal role in reaction-diffusion systems in a study motivated by observations of Amago trout parr-mark pattern formation.

Figure 1

Parr marks on young fish skins of three different species of the Salmonidae.

Figure 2

Amago skin pigmentation patterning during growth development [the approximate time after hatching is given in the captions, and the yellow (light gray) bar in the bottom of each snapshot indicates 1 cm]. The figures show the left- and right-sided patterning profiles during the early stages of the Amago growth development after hatching to the juvenile period, by which time patterning is complete (approximately 6 to 7 months). The fish exhibits an initial patternless state that evolves into faint vertical stripes. It can be observed that stripes evolve into spots with zigzag alignment that increase in number as the fish grows. It must be noted that the fish size at 200 days has not reached the final maturity adult size. It is at this stage that the parr marks remain constant in number. The only changes observed are in the sizes of the spots, which continue to increase.

Figure 3

Top-down view of the patterned state of the juvenile Amago and a sketch of the patterned state on which we highlight the parr marks. A checkerboard pattern is clearly visible.

Figure 4

The red (dark gray) circles represent measurements of the length of an individual fish normalized by its initial length ($\sim 2$ cm) at 0, 20, 60, 90, 160, and 200 days after hatching. The blue (light gray) line represents logistic growth as defined in (1), on the computational time interval $[0,{10}^5]$ corresponding to a real time interval of $[0,200]$ days. It must be noted that the parameters of the logistic function corresponding to experimental growth data for the rest of the fish follows more or less the same logistic growth profile (data not shown).

Figure 5

The portion of the fish that exhibits patterning, shaded in yellow (light gray). To construct an approximation to the surface of the Amago, we assume this region is symmetric about the head-tail axis of the fish and model the region by a series of surfaces with differing curvatures (Figs. 6 and 8).

Figure 6

A planar approximation to the portion of the Amago trout on which patterns form. The length scales, in centimeters, represent the initial planar domain that corresponds to the surface of the Amago soon after hatching.

Figure 7

An example of the mapping $\mathcal{A}$ from a fixed planar computational domain to the surface of a growing elliptic cylinder.

Figure 8

The cylindrical surfaces shaded by the modulus of the mean curvature.

Figure 9

Snapshots, shaded by a threshold algorithm, of the discrete activator concentration ($u_1$), corresponding to the simulation of (4) with reaction kinetics (5) on a planar domain. For parameter and thresholding values see text. The simulations are in agreement with the experimental observations reported in Fig. 2. The first pattern observed is a vertically aligned series of stripe-like parr marks with insertion of new parr marks as the domain grows. The parr marks reorient into a checkerboard configuration as the domain grows further. The number of parr marks visible at each stage of development is also in accordance with the experimental observations reported in Sec. 2. These patterns are relatively independent of the threshold value selected.

Figure 10

Snapshots, shaded by a threshold algorithm, of the discrete activator concentration ($u_1$), corresponding to the simulation of (4) with reaction kinetics (5) on cylindrical surfaces. For parameter and thresholding values, see the text. The timing of the formation of the first patterns is identical to the planar case (Fig. 9). The patterning on the middle three surfaces appears to be very similar to the planar case and approximates well the observed experimental results. A striking result is the preference of stripe-like patterns with vertical alignment on the surfaces with higher curvature on the boundaries (bottom) with spots in zigzag alignment appearing to be preferred on surfaces with higher curvature along the central axis (top). (Note that the surfaces reading from left to right in Fig. 8 correspond to the surfaces reading from top to bottom in each of the subfigures.)