Correspondences Between Parameters in a Reaction-Diffusion Model and Connexin Functions During Zebrafish Stripe Formation

Abstract

Different diffusivities among interacting substances actualize the potential instability of a system. When these elicited instabilities manifest as forms of spatial periodicity, they are called Turing patterns. Simulations using general reaction-diffusion (RD) models demonstrate that pigment patterns on the body trunk of growing fish follow a Turing pattern. Laser ablation experiments performed on zebrafish reveal apparent interactions among pigment cells, which allow for a three-component RD model to be derived. However, the underlying molecular mechanisms responsible for Turing pattern formation in this system remain unknown. A zebrafish mutant with a spotted pattern was found to have a defect in Connexin41.8 (Cx41.8) which, together with Cx39.4, exists in pigment cells and controls pattern formation. Here, molecular-level evidence derived from connexin analyses is linked to the interactions among pigment cells described in previous RD modeling. Channels on pigment cells are generalized as “gates,” and the effects of respective gates were deduced. The model uses partial differential equations (PDEs) to enable numerical and mathematical analyses of characteristics observed in the experiments. Furthermore, the improved PDE model, including nonlinear reaction terms, enables the consideration of the behavior of components realistically.

Full text

Correspondences Between
Parameters in a Reaction-Diffusion
Model and Connexin Functions During
Zebrafish Stripe Formation
Akiko M. Nakamasu *
International Research Organization for Advanced Science and Technologies, Kumamoto University, Kumamoto, Japan
Different diffusivities among interacting substances actualize the potential instability of a
system. When these elicited instabilities manifest as forms of spatial periodicity, they are
called Turing patterns. Simulations using general reaction-diffusion (RD) models
demonstrate that pigment patterns on the body trunk of growing fish follow a Turing
pattern. Laser ablation experiments performed on zebrafish reveal apparent interactions
among pigment cells, which allow for a three-component RD model to be derived.
However, the underlying molecular mechanisms responsible for Turing pattern
formation in this system remain unknown. A zebrafish mutant with a spotted pattern
was found to have a defect in Connexin41.8 (Cx41.8) which, together with Cx39.4, exists
in pigment cells and controls pattern formation. Here, molecular-level evidence derived
from connexin analyses is linked to the interactions among pigment cells described in
previous RD modeling. Channels on pigment cells are generalized as “gates,”and the
effects of respective gates were deduced. The model uses partial differential equations
(PDEs) to enable numerical and mathematical analyses of characteristics observed in the
experiments. Furthermore, the improved PDE model, including nonlinear reaction terms,
enables the consideration of the behavior of components realistically.
Keywords: pattern formation, turing pattern, mathematical model, reaction-diffusion system, connexin, fish
pigmentation
1 INTRODUCTION
In 1952, Alan Turing postulated that two substrates interacting with each other show instability when
they diffuse at different speeds. He explained this diffusion-driven instability by utilizing a linear
reaction-diffusion (RD) model. This model demonstrates that spatial inhomogeneity (i.e., the Turing
pattern) could be generated by such conditions. This relationship is known to generate patterns
though the components remain to be explored.
More than two decades ago, [1] reported that pigment stripes on the bodies of growing marine
angel fish behave as a Turing pattern. The research focus then shifted mainly to zebrafish (Danio
rerio) as a model organism for pattern-formation studies [2–4]. Zebrafish have a pattern of stripes on
their body and fins (Figure 1A). The pattern is generated by three types of pigment cells:
complementarily distributed black melanophores and yellow xanthophores (Figure 1C) plus
iridescent iridophores. Numerous zebrafish pigment-pattern mutants were artificially generated
[5], and the corresponding genes were identified. One of the most important mutants is leopard,
which produces a spotted pattern that is representative of Turing patterns (Figure 1B)[6] identifies
Edited by:
Istvan Lagzi,
Budapest University of Technology
and Economics, Hungary
Reviewed by:
Daishin Ueyama,
Musashino University, Japan
Qingyu Gao,
China University of Mining and
Technology, China
István Szalai,
Eötvös Loránd University, Hungary
*Correspondence:
Akiko M. Nakamasu
nakamasu@kumamoto-u.ac.jp
Specialty section:
This article was submitted to
Physical Chemistry and Chemical
Physics,
a section of the journal
Frontiers in Physics
Received: 30 October 2021
Accepted: 25 November 2021
Published: 18 January 2022
Citation:
Nakamasu AM (2022)
Correspondences Between
Parameters in a Reaction-Diffusion
Model and Connexin Functions During
Zebrafish Stripe Formation.
Front. Phys. 9:805659.
doi: 10.3389/fphy.2021.805659
Frontiers in Physics | www.frontiersin.org January 2022 | Volume 9 | Article 8056591
ORIGINAL RESEARCH
published: 18 January 2022
doi: 10.3389/fphy.2021.805659

connexin41.8 (cx41.8) as the gene responsible for the leopard
mutation [6]. Besides Cx41.8, other connexins, such as Cx39.4,
exist in pigment cells and affect pigment-pattern formation. Six
connexins form a hemi-channel (or “connexon”), which connects
intracellular and extracellular spaces (Figure 1D). Docking of two
hemi-channels from adjacent cells give rise to a gap junction,
which mediates intercellular signal transfer [7]. The minimal
connexin network required to originate a striped pattern was
recently revealed by regulating connexin expression in each
pigment cell [8]. Therefore, these channels are important for
pigment-pattern formation.
Interactions among pigment cells and their molecular
mechanisms involved in pattern formation are summarized in
[9]. However, the molecular mechanisms leading to Turing
instability remain mostly unresolved. Mosaic fish experiments
indicate that both leopard/cx41.8 and jaguar/obelix/kcnj13 genes
FIGURE 1 | Model-based prediction of defects in channels and interactions. (A) Striped wild-type (WT) zebrafish. (B) Spotted leopard mutant zebrafish. (C)
Schematic representation of the relationship between the distribution of pigment cells and the numerical result of the continuous model. (D) Schematic representation of
channels composed of connexin complexes. Hemi-channels are open to the outside of a cell, whereas gap junctio nsform by the docking of hemi-channel connecting to
adjacent cells. (E) Apparent interactions of pigment cells as revealed by laser ablation experiments. (F) Schematic diagram of a three-component PDE model
composed of components Uor V, which correspond to melanophores or xanthophores, respectively, and component w, which represents a highly diffusible molecule.
Interactions are indicated by fine arrows, diffusion coefficients (the motility of the components) are indicated by wide arrows, and corresponding parameters are
indicated. (G) Schematic diagram of the effect of channels on pigment cells according to the mathematical model. Parameters related to the function of each gate are
indicated by hatchings.
Frontiers in Physics | www.frontiersin.org January 2022 | Volume 9 | Article 8056592
Nakamasu PDF Model for Fish Pigmentation

are required for segregation of melanophores and xanthophores.
Such segregation is proposed to involve local interactions
between adjacent pigment cells [10]. Xanthophore ablation
using a temperature-sensitive csf1ra allele led to the gradual
death of melanophores in both the body trunk and fins of
adult fish [11]. Accordingly, melanophore survival requires
continuous signaling from xanthophores. Laser ablation of
stripe and interstripe areas has revealed the mutual
interactions between melanophores and xanthophores [12].
The interactions comply with the requirements for Turing
pattern formation (Figure 1E). Specifically, both types of
pigment cells activate their own types at a single-cell distance
(short range) by inhibitions of other types and then inhibit
their own types beyond the width of the stripe (long range).
The difference in reaction distances achieves the “local
activation and lateral inhibition”condition needed for
pattern formations [13]. It should be noted, however, that
the distinction between iridophores and xanthophores is
sometimes unclear in those experiments.
To explain the opposing actions at long vs. short distance, a
model that includes a highly diffusible molecule (i.e., long-range
factor) and two cells (regarded as short-range factors with low
diffusivity) was constructed [12]. This three-component RD
model with its linear reaction terms and upper and lower
limits describes the apparent interactions obtained
experimentally. Then, the different diffusibilities and the
interactions in the model achieved diffusion-driven instability
(Turing instability).
Further investigations reveal that the interactions are
mediated by cell projections [14]. The interaction mediated
by gap junctions on the tip of the projection is considered to be
a long-range effect observed in the previous experiments [15].
The researchers mention the possibility that the pattern
formation might not require actual diffusion. Later, a
Turing model based on an integral kernel was suggested
[16] though the link between parameter and molecular
function was ambiguous. Most other models for pigment-
pattern formation are based on interactions at a cellular
level. These models implement different effects depending
on the distance from each pigment cell by agent-based
models [17,18] and by minimal lattice models [19,20].
Several attempts were made to explain the observed
patterns in zebrafish mutants by a general Turing model
[21,22]; however, they were not supported experimentally
even though there are several paths to cause the expected
pattern changes in mutants.
Here, the interactions in a three-component model,
including a hypothetical highly diffusible factor, are
developed to attempt to link the molecular functions of
connexins in zebrafish. Channels thought tobeimportantfor
pattern formation are generalized as “gates”of pigment cells.
These gates enable transport of the diffusible molecule across
the membrane. The parameters affected by each gate are
deduced; then, the effects on pattern selection and size are
analyzed. Finally, the model is improved to an analogous
model with nonlinear terms. These models together enable
reasonable explanations of detailed behavior of the
components that relate to the pattern formation.
2 MATERIALS AND METHODS
2.1 Numerical Simulations
For the linear model, duwas increased from 0 to 0.2 within
limits of the reaction term along the x-axis (Figures 4A,
D–H). It can change the distance between the equilibrium
point of u and the upper limitation [23] without shifting the
maximum of the limit, and the parameters pvw and cwv were
decreased linearly from 1 to 0 (Figures 4B,D–H)for
investigation of the effect of gates on each cell. For short-
range effects, iuv and ivu were decreased linearly from 1 to 0.6
in Figure 4C. Accordingly, the arbitrary parameter set
generating stripes (wild type) was placed in the right top
of the phase plane (Figures 4D–I). Partial differential
equations (PDEs) were calculated with 20,000 and 40,000
iterations with dt 0.1anddt 0.05 in fields sized xl 56.25,
yl 225 with dx 0.75, and xl yl 200 with ds 0.5for
Figures 4A–Cand Figures 4D–H, respectively. These
conditions satisfy each CLF condition. In Figure 4I,to
investigate the simultaneous gap-junction effects with
hemi-channels sd(x, y)1−0.002 max {x, y},
ud(x)0.0004x,andvd(y)1−0.002ywere utilized in
the field sized xy300(dx 0.75)and dt 0.1for
20,000 iterations.
For the nonlinear improved model with nonlinear terms, the
parameters du,iuv, and ivu were decreased linearly from 1 to 0.6.
cwv and pvw were decreased linearly from 0.5 to 0.1 in Figures
5A–C and decreased by sd(x, y)1−0.002 max {x, y}
simultaneously in Figure 5D. Accordingly, the arbitrary
parameter set generating stripes (wild type) was placed in the
right top of the phase plane. PDEs were calculated in fields sized
xl 10, yl 40 in Figures 5A–Cand xl yl 50 with ds 0.25
in Figure 5D. Then, after 500,000 iterations calculated with
dt 0.01, we obtained the result.
Calculations were performed in the language Full BASIC ver.
8.1 with no-diffusion boundary conditions with difference
calculus; then, results are shown as density plots of u.
Parameters utilized in this study are summarized in Table 1.
2.2 Quantification of Simulated Color
Patterns
Color pattern complexity and overall tone were quantified
from binarized images using ImageJ as described in [24].
Briefly, the pattern simplicity score (PSS) is defined as the
area weighted mean isoperimetric quotient of the contours
extracted from each image. The overall color tone (OCT) of a
pattern is defined and calculated as the ratio of white pixels in
the binarized image. Analyzed images were prepared by the
quaternary connection of a numerical result (100 ×100
individual fields with periodic boundary conditions) of u
in each parameter.
Frontiers in Physics | www.frontiersin.org January 2022 | Volume 9 | Article 8056593
Nakamasu PDF Model for Fish Pigmentation

3 RESULTS
3.1 Linear RD Model for Pigment-Pattern
Formation
Previous laser ablation experiments reveal that the presence of
mutual interactions between two types of pigment cells are
necessary to generate Turing patterns (Figure 1E). Briefly, the
density of melanophore existing and newly generated in a stripe
decreases when the xanthophores in adjacent interstripes are
ablated. On the other hand, that of xanthophore was not
drastically changed. Then, two types of pigment cell inhibit
each other at a one-cell distance even though the inhibition
from xanthophore is inapparent until melanophores in
adjacent stripes are eliminated. A mathematical model is
derived from these apparent interactions in [12] although the
details of the relationship between the experimental results and
the model are not described. This model is based on the following
set of RD equations:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
zu
ztDu2u+f(v,w)−duu
zv
ztDv2v+g(u,w)−dvv
zw
ztDw2w+h(u,v)−dww
. (1)
Here the alternative distribution of two types of pigment cells
(Figure 1C) is expressed by two factors (U, V)of the three
components. Then, uand vare each volume (viability). The
numerical simulation of this model results in a Turing pattern in
which uand vare distributed with antiphase, and a concentration
of third factor W(w)presents peaks synchronized with u
(Figure 2). It should be noted that cell divisions of
differentiated melanophores contribute only minimally to the
pigment-pattern formation in fish (Figure 1F). Therefore, the
number of melanophores is changed by 1) the supply of new cells
from randomly scattered precursor cells, 2) the death of existing
pigment cells, or 3) the migration from a position close to the skin
surface [25]. In the case of melanophores, it is known that cell
movements and cell deaths are complementary to each other [26].
Even though they are inhibited, xanthophores are found in the
stripe region, where they exist with a pale color [27–29].
Xanthophores do not move actively in vivo as may be the case
for iridophores. As detailed in Figure 1F, motilities of the cells are
approximated by small diffusion coefficients (D). The rapidly
diffusing factor Wis assumed to have a large diffusion coefficient
1 in the outer region of the cells based on the results of electro-
physiological experiments [30–32].
In the reaction, rather than it should be called an interaction,
formulae, the dimension-less parameters are chosen arbitrarily
from the sets that bring diffusion-driven instability. They are
positive constants as shown in Table 1.Eachformulais
composed of a set of linear terms with upper and lower
limits as utilized in the two-component system mentioned by
[1]asfollows:
TABLE 1 | Parameter set utilized in this paper.
Du,Dvduiuv iuw suivu dvpvw svDwpwu cwv dwswfmax,gmax h
max
Figure 2 0.01 −)0.01 0.05 0.05 0.05 0.05 0.01 0.05 0 1 0.05 0.07 0.05 0.02 0.01 0.05
Figures 4A–D
Figure 4E 0.01 −)0.01 0.04 0.05 0.05 0.04 0.01 0.05 0 1 0.05 0.07 0.05 0.02 0.01 0.05
Figure 4F 0.01 −)0.01 0.03 0.05 0.05 0.03 0.01 0.05 0 1 0.05 0.07 0.05 0.02 0.01 0.05
Figure 4G 0.01 −)0.01 0.02 0.05 0.05 0.02 0.01 0.05 0 1 0.05 0.07 0.05 0.02 0.01 0.05
Figure 4H 0.01 −)0.01 0.01 0.05 0.05 0.01 0.01 0.05 0 1 0.05 0.07 0.05 0.02 0.01 0.05
Figures 5A–D0.01 +)1 1 1 0.2 1 0.1 1 0.5 1 1 1 1 0.0 -——
FIGURE 2 | Turing pattern obtained by a reaction-diffusion model in Eqs
1,2.(A,B). Calculation results of the model had linear terms with limits. (A)
One and (B) two dimensions. Results indicate Turing patterns in which the
variables uand vare distributed complementarily, and the highly
diffusible molecule Wpeaks at high u.
Frontiers in Physics | www.frontiersin.org January 2022 | Volume 9 | Article 8056594
Nakamasu PDF Model for Fish Pigmentation

⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
f(v,w)⎛
⎜
⎝0;−iuvv−iuw w+su<0
−iuvv−iuw w+su;0<−iuvv−iuw w+su<fmax
fmax;−iuv v−iuw w+su>fmax ⎞
⎟
⎠
g(u,w)⎛
⎜
⎝0;−ivuu+pvw w+sv<0
−ivuu+pvw w+sv;0<−ivu u+pvww+sv<gmax
gmax;−ivu u+pvw w+sv>gmax ⎞
⎟
⎠
h(u,v)⎛
⎜
⎝0;p
wuu−cwv v+sw<0
pwuu−cwv v+sw;0<pwu u−cwv v+sw<
hmax;p
wuu−cwv v+sw>hmax
hmax⎞
⎟
⎠
.
(2)
These equations simply indicate negative or positive
interactions among two cells and molecules by the coefficients
with different signs. They are derived from the interaction
network obtained by the experimental results in [12]. The cells
are assumed to be inhibiting mutually (−iuv,−ivu ), and then Wis
assumed to be produced by U(pwu)and consumed by V(−cwv ).
Accordingly, Wis assumed to inhibit pigment cells of the
producer (−iuw)and produce (or preserve) the consumer
(pvw). The self-coupling parameters −du,−dv,−dw
corresponded to degradation (or death) coefficients, whereas
s’s represent constants related to the supply (also called
“support sustainability”) of each component. The producer U
activates Vbut then inhibits itself at long range via W.By
consuming W,Valso indirectly inhibits itself but then is
activating Uby double inhibition at long range. As a result, U
and Vexhibit no difference in apparent interactions, making it
difficult to identify which factor corresponds to which cell type.
Besides melanophores, xanthophores also show self-inhibition at
long range. In laser ablation experiments, melanophore
elimination in adjacent stripes causes pale-colored
xanthophores in an interstripe region. Therefore, the pale
color reflects xanthophore inhibition even without a change in
cell number.
The three-component model in (1) is somewhat complex
though it can be roughly regarded as a combination of two-
component systems originally suggested by [33] as follows. For
ease of mathematical analyses, I use the following two-
component system that shares a component with high-diffusivity.
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
zx
ztfxx+fyy+Dx2x
zy
ztgxx+gyy+Dy2y
. (3)
There are two different cases that bring diffusion-driven
instabilities, i.e., activator–inhibitor type (Figure 3A)or
activator–substrate–depletion type (Figure 3B), characterized
by different signs of the parameters [34,35]. Both conditions
are included in the three-component model sharing with the
high-diffusible component (Figure 3C). Therefore, each
parameter in the model can be regarded as part of a two-
component system. The first and second reaction Eq. 2
include mutual inhibitions (−iuvv, −ivu u), each of which
corresponds to the respective self-activation (fx>0 for x)
though it cannot be realized without each partner. Recent
experiments reveal that xanthophores are generated from
division of other xanthophores [29]. Therefore, at least
one self-reaction parameter; −dfor uor vin Eq. 2 cannot
be a degradation coefficient; i.e., it might be a nonnegative
parameter.
3.2 Effects of the Parameters on the
Component Proportion and the
Characteristic Wavelength of a Pattern.
Variation in patterns observed in most zebrafish mutants is
explained by changes in the types and sizes of patterns. The
former is defined by pattern selection [23,36] and manifests as a
general variation of the Turing pattern from spots to stripes to
reverse spots. The latter is dictated by the characteristic
wavelength of the pattern [37]. The following sections
analytically describe the effect of parameters in the model
(1)–(2) on these characteristics.
FIGURE 3 | Parameter correspondences to two-component systems
and effects on pattern size. (A,B) Two conditions that generate a Turing
pattern: activator–inhibitor type (A) or activator–substrate–depletion type (B)
with different signs of reaction parameters. (C) Correspondence
between parameters in the present tree-component model and the two-
component systems. Interactions included in conditions (A) and (B) are
enclosed by solid and dashed polygons, respectively. (D) Matrix forms of
reaction terms corresponding to the two-component system: the filled
squares indicate the parameter that increases pattern size when the absolute
value is decreased, and the open squares denote the opposite; the solid
squares correspond to condition (A), and the dashed ones correspond to
condition (B).
Frontiers in Physics | www.frontiersin.org January 2022 | Volume 9 | Article 8056595
Nakamasu PDF Model for Fish Pigmentation

3.2.1 Effects of Parameter on Pattern Size
The characteristic wavelength of a pattern can be analytically
obtained from the dispersion relation of linear stability [37]. The
wavelength of a two-component RD system in (3) is given by the
following equations of which derivation was mentioned in detail
in [37]:
2π
kmax


DxDy
Dy−Dx
Dx+Dy
−fygx
−
DxDy
fx−gy

. (4)
kmax is a preferable wave number in a system. As mentioned, each
parameter in the model can be regarded as a two-component
system (Figure 3C). The mutual inhibitions (−iuvv, −ivu u)in the
three-component system correspond to self-activation though
they are inversely related; i.e., the sign of the parameters is
different from fxin two-component systems. The increase in
the absolute values of the parameters reinforces the self-
activation. At the same time, the effects of −duand −dvare
the opposite of fx. The effect on pattern size when the absolute
value of each parameter is decreased (i.e., the decline of each
interaction) is shown in Figure 3D and indicates correspondence
with two-component systems (Figures 3A,B).
3.2.2 Effects of Parameter on Pattern Selection
In zebrafish, pattern selection is determined mainly by the
proportion of two types of pigment cells with complementary
distribution. The relative position of the equilibrium from the
limits of the reaction terms provides an index for pattern selection
[23] as that is the origin of diffusion-driven instability.
Considering the differential equations, the decrease in each
absolute value of parameter (i.e., the decline of each interaction)
with a positive effect decreases the population volume of
respective cells; then that of the parameter with a negative
effect increases the population of the respective cells. Here, the
component Wwith high diffusivity represses uand promotes v;
therefore, the decrease in the positive parameter in the differential
equations of wincreases uand decreases v, respectively. The
opposite can apply to negative parameters. From this aspect,
however, it is difficult to refer about the combined effect of
parameters with different signs.
3.3 Correspondence Between the
Mathematical Model and Connexin Defects
in Zebrafish Estimated from Molecular
Function
Next, correspondence between this model and molecular
functions is assumed (Figure 1G). In zebrafish, leopard
mutants are known to have an aberrant pigment pattern,
whereby stripes are changed to spots [6](Figures 1A,B). The
gene responsible for the leopard mutation is a connexin cx41.8,
which encodes a four trans-membrane connexin protein.
Additionally, mutation of connexin cx39.4 results in wavy
stripes. Recent analyses of connexin activity reveal different
functions associated with hemi-channels and gap junctions
[30,31]. Hemi-channels are open to the extracellular
environment, whereas gap junctions form connections between
cells to allow the exchange of small molecules (Figure 1D).
Accordingly, connexins may be involved in both long- and
short-range interactions. These channels may function as gates
for the transport of molecule Wacross the membrane
(Figure 1G). Accordingly, producer Uproduces W, which
then diffuses outside the cell into the extracellular space via
some kind of gate. Therefore, gate defects affect survival of the
producer by preventing the release of harmful W(i.e., duof −duis
increased). I consider two other possibilities, i.e., the effect of the
gate defect on Dwand/or iuw.IfDwis changed, Woutside of the
cells is also e affected. A u-dependent decrease in Dwmight be
more appropriate for the assumption of enclosed Wthough it
gives difficulties in the analytical approach and both finally result
in an increase in the death of U. Then, the degree of harmful effect
on Uby the same concentration of wdoes not change. Therefore,
Dwand iuw are not affected. whas peaks with the peaks of
producer U, and gates on cells assume passive effects on W
movements. This explains why pwu might not be affected at least
directly. Consumer Vis assumed to incorporate beneficial Winto
the cell across the gates and then to consume it, indicating that the
gate defects decrease the parameters cvw of −cvw and pwv. Both
parameters are assumed to be related to intracellular events;
therefore, higher wis needed for the same rate of
consumption and Vproduction compared with the case of
intact hemi-channels. At the same time, parameters for mutual
inhibitions iuv and ivu seem to be decreased by the leopard
mutation [38] through gap junctions (combination of gates) at
short range. They are simultaneously affected, linking with the
hemi-channel defect on the corresponding cell. In Figure 1G,
these parameters linked to different gates are indicated by
different hatchings.
3.4 Comparisons of Results Obtained by
Simulation and Experiments.
First, the independent effects of the gate on each cell were
analyzed numerically. When an arbitrary stripe is set as a
starting point, only gates that open to the outside on Uand V
cells were removed along the x-andy-axes. Numerical
simulation of this model yielded a spot pattern of uin the
case in which gates on Ucellshavedefectsasexpectedbythe
sign of the parameter (Figure 4A). Reversed uspots are
yielded in Vcell defects (Figure 4B) though the change is
not strong because it includes opposite effects on Vvolume.
PSS increases in both cases, and then OCT are decreased and
increased by respective defects. That is, the asymmetry of
changes in pattern selection can be observed by the removals
of gates on respective cell types.
Defects on short-range inhibition do not have drastic effects
on pattern selection though the pattern does finally disappear
(Figure 4C). On the 2-D plane, the stripe region is recessive
together with the defect in short-range effects by gap junctions
(Figures 4D–H,J) though the tendency to shift the pattern
selection is not changed. In Figure 4I, short-range effects are
simultaneously decreased by respective xor yvalues that link
with Uor Vdefects as shown in the right panel in Figure 4J. This
Frontiers in Physics | www.frontiersin.org January 2022 | Volume 9 | Article 8056596
Nakamasu PDF Model for Fish Pigmentation

FIGURE 4 | Numerical results of gate effects and comparison with experiments. (A–I) Numerical results obtained from a model of linear reaction terms with limits.
Pattern shifts were observed when the effects of gate defects on both cells were changed. Defects of gates on Uand Vcells and between the cells become larger along
the x−,y−,andz(or x,y)−axes, respectively. (A–C) Each defect was analyzed independently with quantification of patterns. (A) Udefect, (B) Vdefect, and (C) short-range
defects. PPS and OCT were obtained quantitatively. The short-range effects by gap junctions were eliminated from arbitral wild-type condition (D) 0.05 to (E) 0.04,
(F) 0.03, (G) 0.02, and (H) 0.01. (I) Numerical result of linear terms with limits in which the short-range effects were spontaneously changed along the x−and y−axes with
increasing udand decreasing vd. Each small letterindicates the corresponding fish withconnexin conditions. (J) Schematics of this analyses utilizing the present linear model.
Udefect, (B) Vdefect, and (C) short-range defects were changed along with the axes in each panel as shown in the figures. (K) Pigmentation pattern on connexin-
manipulated fish slightly changed from [8], with permission to use the figures from Dr. M. Watanabe. Gray flamed fish had correspondence to numerical results.
Frontiers in Physics | www.frontiersin.org January 2022 | Volume 9 | Article 8056597
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also keeps the same tendency to shift the pattern selection with
individual cases.
The results mostly consist of the positive effects of connexin
additions to WKO that increase the respective pigment cells [8].
However, experimental eliminations of the gate on each wild-type
pigment cell lead to an increase in the rate of respective pigment
cells; melanophore defects generate net (or rather wavy stripe)
patterns of melanophores, and the xanthophore defects result in
dot patterns of melanophores (i.e., net patterns of xanthophore).
Therefore, the simulation results are partly inconsistent with
experimental results in [8] in which the effects of connexins
on each or both pigment cells are investigated in detail
(Figure 4K).
From the electro-physiological experiments in [30]; each
type of connexin can be considered to have different
strengths of the (hemi-) channel functions on the two
types of cells. Deduced patterns of respective transgenic
fish are indicated by small letters on the phase plane in
Figure 4I. Even though the differences in the strengths of
channels are taken into account, the removal of hemi-
channels on wild-type xanthophores tends to increase the
proportion of xanthophores; then a faint increase of
proportion in melanophore is brought in the case of
connexinonmelanophores(Figure 4K). If cell Uis a
melanophore, other than that the gray-framed patterns
shown in Figure 4K do not seem to correspond to
Figure 4I, all of the experimentally obtained patterns
exist in the simulation.
From the analyses of wavelength mentioned above, defects to
the gates on cell U(increasing in duof −du) or cell V(decreasing
in pvw and cwv) cause a decrease or an increase in pattern size,
respectively. In numerical simulations, the pattern size tends to be
decreased and increased by the hemi-channel defects on both U
and Vcells as expected (Figures 3A,B). The characteristic
thinning of ustripes and widening of vinterstripes are
observed in the simulation of U-cell defects though it is not
explained by the analyses. From the characteristic thinning of the
ustripe, each cell corresponding to one of two short-range factors
may be deduced. However, the thinning is observed on
melanophore stripe in the case of a defect in the xanthophore.
Inconsistent with the Udefect, the melanophore defect tends to
result in wide melanophore stripes in the experiment.
3.5 Improvements to the Model with
Nonlinear Terms.
To describe the detailed behavior of the components in the
system, the model is changed to a model including nonlinear
terms as shown in (5).
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
zu
ztD∇2u−iuvuv −iuw uw +duu+su
1+u+v
zv
ztD∇2v−ivuvu +pvw vw −dvv+sv
1+u+v
zw
zt∇2w+pwuu−cwv wv −dww+sw
. (5)
The qualitative relationships for pattern formation shown in
Figure 1F are not changed from model (1) and (2). That is,
mutual inhibitions between Uand V(−iuvuv, −ivuvu)are
assumed; then, substance Wwith high diffusivity is assumed to
inhibit the producer U(−iuwuw)and to promote consumer
V(pvwvw). Considering the interactions between different types
of cells and between cells and molecules, the inhibitions were given
by multiplication terms (e.g., the inhibition of Uby Wwas
described as uw and so on). These multiple terms are based on
the description of the second order reaction in the chemical kinetic
equation ordimer reactions by [39]; it enables only limited reaction
by the contacts between the components. Wis needed for
maintenance of cell V, so this reaction is also given by a
multiplication term of their volume. On the other hand,
production of Wby Uonly occurs u-dependently, and
degradation (and death) is also w-dependent. Therefore, those
terms are linearly related to each component. This model as an
example of possible improvements also generated Turing patterns
(Figure 5). These improvements can identify the functions on
existing cells or newly differentiating cells. Then, new generations
of pigment cells occur only with eliminations of existing cells [40].
Therefore, mature cells would inhibit the new generation of cells.
The sign of duseemed preferably to be positive for the starting
point of pattern selection corresponding to zebrafish, i.e., the start
from stripe. Even though the sign wasopposite to the linear model,
total uchange may become minus with relation to other
components (i.e., −iuvv−iuww+du<0 in the deformed reaction
terms ((−iuvv−iuww+du)u+su/(1+u+v)).Therefore,the
self-productivity can be small enough to agree with the low
proliferation rate of melanophores. Similarly, concerns about
the self-productivity of xanthophores mentioned above, zv/ztin
(Eq. 5) already have self-productivity by the multiplication term vw
regardless of the sign of dv(i.e., −ivuu+pvw w−dv>0indeformed
reaction terms (−ivuu+pvw w−dv)v+sv/(1+u+v)). The sign
of dvcan be inverted though the change is not expected to
substantially affect pattern characteristics.
Next, numerical calculations of the nonlinear model are
performed to consider the condition in which respective gates
on Uand Vcells and both-gate defects are added (Figures 5A–C).
Again, numerical results consistent with the linear model can be
obtained from an arbitrary parameter set generating a stripe
pattern. A biased pattern shift can also be obtained by
simulations, partly corresponding to connexin-mutation
experiments. When gates on the Ucell are deleted, it results
in a udot pattern (Figure 5A). Simultaneous decreases in pvw and
cwv by increasing the gate defect on Vcells generate a net of u
though not so drastic (Figure 5B). A defect in the gap junction by
decreasing ivu and iuv has a minimal effect though the stripe
region became recessive with a combination of defects on each
outer gate (Figures 5C,D). The thinning of the upopulation is
not clear because of the thin stripe at the start.
4 DISCUSSION
In the present study, to confirm the potency of diffusion-driven
instability in determining fish pigmentation patterns, channels on
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Nakamasu PDF Model for Fish Pigmentation

pigment cells are generalized as gates (Figure 1). The three-
component RD model (1) is shown to be composed of two-
component RD systems bringing diffusion-driven instabilities
(Figures 2,3). The proposed qualitative models help to
understand the relationship between pigment cells as well as
between cells and molecules even when their identity is unknown.
The terms of the theoretical model are connected with the
functions of each channel on different cells. Parameters
affected by the gate defects were deduced (Figure 1G); then,
the effects of such defects are simulated from a parameter set that
generates an arbitrary stripe as a benchmark (Figures 4,5). Then,
dots and thinning of the Ucell population can be obtained by a
defect of the gate on it in numerical simulations. The identities of
melanophores and xanthophores are deduced from the change of
pattern selection and the wavelengths though the identify of
important substance Wis still missing. Then, improvements
to the nonlinear model enable a description of the detailed
behaviors of components that are related to pattern formation.
Though the numerical analyses cannot strictly explain the pattern
obtained by experimental manipulation of connexins by [8]; the
present study can help to interpret the mechanism underlying the
leopard mutation as a Turing system.
The determination of the signs of self-reaction terms for Uand
Vis difficult. The signs of duprefer to invert for a desirable range
of pattern selection starting from the stripe in the improved
model though the total changes for U(melanophore)
correspond to experimental observations. The increase in
existing melanophores brought by positive duare visible by
the laser ablation of adjacent xanthophores or deletion of
stripes in the experiments in [12]. Each manipulation
decreases −iuvvu or −iuw wu inhibitions, respectively. On the
other hand, the self-productivity of xanthophores in the
nonlinear model can be achieved even with the negative
coefficient of self-reaction. The combination of two cases of
diffusion-driven instability in the three-component model
indicates the capacity to make a pattern without melanophores
by the self-productivity of xanthophores if sufficient Wis added
externally. On the other hand, if melanophores have strong self-
productivity, they are also able to make a pattern without
xanthophores by the removal of extra W.
Connexins are related to both hemi-channels and gap
junctions. Hemi-channels are considered less important in
physiology although it was recently revealed that the aberrant
activity of hemi-channels can change the proportion of vertebrae
[31] and are related to the pigment-pattern formation in zebrafish
[30,32]. Laser ablation experiments show that the interactions
between xanthophores and melanophores differ depending on
the distance. In the present model, the highly diffusible molecule
Whas a positive effect on V. Hence, the inhibition of Vvia gap
junctions is inconsistent. Outflow of harmful Wfrom via gap
junctions is also inconsistent with the inhibition. Furthermore,
generation of a new gap junction between two cells takes more
than 30 min [41] (Watanabe personal communication). This
elicits a different signal transduction cascade as cell
depolarization [42] and incorporation of functions for
molecules other than connexin should be envisioned. Then,
the inverted function may be derived from an observed
rectified current in the gap junction. It is observed that quail
melanocytes interact with each other via filopodia in vivo and
in vitro [43]. Therefore, gap junctions may have functions on
interactions not only between other types of cells but among the
same populations though the obtained simulation results
included several collisions with experimental results. Similar
discords are also in experiments. The results of [8] indicate
that either Cx41.8 or Cx39.4 is needed on melanophores; then,
Cx41.8 is necessary and sufficient on xanthophore for (stripe)
pattern formation. As plotted in Figure 6A, the proportion of
pigment cells is drastically changed in the case of the
FIGURE 5 | Numerical results of gate effects by nonlinear model. (A–D) Numerical results obtained from a model with nonlinear reaction terms. Pattern shifts were
observed when the effects of gate defects on both cells were changed. (A) U defect, (B) Vdefect, and (C) short-range defects, and between the cells, became larger
along the x−,y−, and z(or x,y)-axes, respectively. Pattern quantification by PSS and OCT are shown on the right. Then, the short-range inhibitions were decreased along
both axes, spontaneously. All results are shown as density plots of u.
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Nakamasu PDF Model for Fish Pigmentation

manipulation of the gate on xanthophores though asymmetry of
change in color tones of simulated pattern in Figures 4A–C
indicate the gate on melanophores is effective. Shuffle of mutant
fish further supports the numerical results shown in Figure 6B.In
this case, expected “Cx41.8”manipulations show the “necessary
and sufficient”trait on melanophores.
Similar effects of connexin mutations (i.e., shift from stripe to dots)
can be observed on both the body and fins. Because the fins lack
iridophores, the effect on the pigment pattern formation depends on
the relationship between melanophores and xanthophores. Even
though pattern formation is achieved by the two types of cells,
details on the role of iridophores in cellular interactions are
revealed [10]. The evaluation of the iridophore function in similar
modeling is also possible and should be attempted.
Using such a model with PDEs will lead to various
mathematical analyses. For example, pattern size was
mathematically analyzed with regard the model as
combination of two-component systems here. This method
cannot yet describe the independent pattern sizes of each type
of pigment cell that are observed experimentally [22] and
predicted numerically in this paper. Therefore, more
sophisticated analyses are required in the future.
DATA AVAILABILITY STATEMENT
The raw data supporting the conclusion of this article will be
made available by the authors, without undue reservation.
FIGURE 6 | Shuffle of mutant fish that gives accordance with numerical results. (A) A scatterplot of two types of pigment cells. Stars and dots indicate shifts of
proportion of pigment cells caused by experimental manipulations of connexin by [8]. However, most data can be obtained only about melanophores, so changes in
melanophore number are indicated by broken lines. Simple broken lines indicate the addition of connexins to double knockout (WKO); black ones are “on melano phore,”
and gray ones are “on xanthophore,”and broken lines with solid lines on the back denote deletion of connexon from WT; black dot on gray corresponds to “from
melanophore”and gray dot on black “from xanthophore.”Solid lines indicate the proportion of WT. Open stars and filled ones indicate the values of WKO and WT,
respectively. (B) Mutant fish are shuffled for accordance with numerical results in Figure 3I. Each small letter indicates the corresponding fish with simulation in the
aspect of pattern selection.
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Nakamasu PDF Model for Fish Pigmentation

AUTHOR CONTRIBUTIONS
The author confirms being the sole contributor of this work and
has approved it for publication.
FUNDING
This research was supported from Grant-in-Aid for Scientific
Research on Innovative Areas (The Japan Society for the
Promotion of Science), Periodicity and its modulation in plant
No. 20H05421 and Research grant from Shimadzu Science
Foundation.
ACKNOWLEDGMENTS
The author thanks Masakatsu Watanabe and Masafumi Inaba for
stimulating discussions and important comments as well as members
of the Higaki lab for providing a suitable environment to concentrate
on this investigation. I also thank Editage (www.editage.com) for
English language editing and the IROAST Proofreading/Publication
Support Program. I would like to thank to S. Kondo.
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