Team:Kyoto/ProjectTuring

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=Turing Model<br>-the problems between wet and dry-=
=Turing Model<br>-the problems between wet and dry-=
==Introduction==
==Introduction==
-
On the Earth, there are various animals which have various patterns on their skin. The formation mechanism of this pattern have not been explained by any verified theories, although many hypothesis are proposed. Among these hypothesis, there is a model pattern called Turing pattern proposed by A. Turing the famous mathematician *1. S. Kondo *2 and some other researchers *3 suggests that some creatures’ pattern can be explained by Turing’s model. Here we will explain how the Turing pattern is expressed by his model step by step.<br>
+
On Earth, there are various animals which have various patterns on their skin. The mechanism of this pattern formation has not been explained by any valid theories yet, although many hypothesis has been proposed. Among these hypothesis, there is a model pattern called Turing pattern proposed by A. Turing, a famous mathematician *1. S. Kondo *2 and some other researchers *3 suggest that some creatures’ pattern can be explained by Turing’s model. Here we will step by step explain how this Turing pattern is expressed by his model.<br>
-
[[File:IGKU0002.png|300px]]<br>
+
[[File:IGKU0002.png|300px]][[File:sakana.png|400px]]<br>
-
Let’s take a look on simple hypothetical pattern formed by just two colors. Creatures’ epidermal pattern exists on the cells. Let’s assume that the pattern is formed by cells in different state &alpha; and &beta; for example. Cell in state &alpha; expresses color 1 and changes close cell in state &beta; into state &alpha;. Cell in state &beta; expresses color 2 and change close cell in state &alpha; into state &beta;, and remote cell in state &beta; into cell in state &alpha;. For convenience, hereafter we call the cell in the state of &alpha; by {&alpha;} and cell in the state of &beta; by {&beta;}.<br>
+
Let’s take a look on a simple hypothetical pattern formed by just two colors. Creatures’ epidermal pattern is expressed on the cells. Let’s assume that the pattern is formed by cells in different state α and β for example. A cell in state α expresses color 1 and changes close cell in state β into state α. Another Cell in state β expresses color 2 and changes close cells in state α into state β, and remote cells in state β into state α. For convenience, hereafter we call the cell in state α {α}, and cell in the state β {β}.<br>
[[File:IGKU0003.png|200px]][[File:IGKU0004.png|400px]]<br>
[[File:IGKU0003.png|200px]][[File:IGKU0004.png|400px]]<br>
-
Then, we will take a look on the system that two cells {&alpha;} and {&beta;} are existing uniformly and both of the cells are in the equilibrium state of interaction. Now suppose that the density of {&alpha;} and {&beta;} fluctuated in somewhere in the system. Assume that the density of cell {&beta;} increases like the cells in center of figure 1. At first, the center {&beta;} changes close {&alpha;} into {&beta;}. And next same {&beta;} changes remote {&beta;} into {&alpha;}. Then remote {&alpha;} changes close {&beta;} into {&alpha;}. The pattern is formed as this interaction continues.<br>
+
Now, we will take a look at the system where two cells {α} and {β} exist uniformly and both of the cells are in the equilibrium state of interaction. Now suppose that the density of {α} and {β} fluctuated somewhere in the system. Assume that the density of cell {β} increases as shown in the center of figure 1. First, {β} in the center changes the neighboring {α} into {β}. And next the same {β} changes the remote {β} into {α}. Then remote {α} changes neighboring {β} into {α}. The pattern forms as this interaction continues<br>
-
 
+
[[File:stripeform.gif]]<br>
-
Like this, a striped pattern is formed from close-and-remote interaction between two states of cell. Seeing this close-and-remote interaction separately, close interaction can be explained as positive feedback reaction in the aspects of polarization. Conversely, remote interaction can be explained as negative feedback reaction.<br>
+
Like this model, a striped pattern is formed by close-and-remote interactions between two states of cells. When we look at this close-and-remote interaction separately, close interaction can be explained as positive feedback reaction in the aspects of polarization. Conversely, the remote interaction can be explained by negative feedback reaction<br>
[[File:IGKU0006.png|230px]][[File:IGKU0001.png|300px]]<br>
[[File:IGKU0006.png|230px]][[File:IGKU0001.png|300px]]<br>
-
Diffusing substances such as proteins secreted from the cells determine the characters of the cells. It seems that the characters which changes close or remote cells (i.e. &alpha; and &beta;) are function of these diffusing substances. In other words, it can be said that {&alpha;} and {&beda;} secretes different diffusing substances, and the substances lead to close interaction (positive feedback) and remote interaction (negative feedback). Therefore, the pattern formation can be said to be formed by interaction between diffusible substances, as well as cell-cell interaction.<br>
+
Diffusing substances such as proteins secreted from the cells determine the characters of the cells. It seems that the characters which changes close or remote cells (i.e. &alpha; and &beta;) are function of these diffusing substances. In other words, it can be said that {&alpha;} and {&beta;} secretes different diffusing substances, and the substances lead to close interaction (positive feedback) and remote interaction (negative feedback). Therefore, the pattern formation can be said to be formed by interaction between diffusible substances, as well as cell-cell interaction.<br>
[[File:IGKU0008.png|200px]]<br>
[[File:IGKU0008.png|200px]]<br>
-
Then let’s consider about this interactions between diffusible substances in simplified model. Living organism’s body surface consists of cells shaped and sized ununiformly, therefore it is easier to understand if you assume that the body surface is a plane and consists of square cell-units sized uniformly. In this model, we can set diffusible substances which are secreted by {&alpha;} and {&beta;}, and they increase and decrease under the influence of interactions. And then, they are substances which has the same characteristic of cell {&alpha;} and {&beta;} (substances lead to close interaction (positive feedback) and remote interaction (negative feedback)), as a causative agent of the pattern formation on this model surfaces.<br>
+
Then let’s consider about this interactions between diffusible substances in simplified model. Living organism’s body surface consists of cells shaped and sized ununiformly, therefore it is easier to understand if you assume that the body surface is a plane and consists of square cell-units sized uniformly. In this model, we can set diffusible substances which are secreted by {&alpha;} and {&beta;}, and they increase and decrease under the influence of interactions. And then, these substances have the same characteristics of cell {&alpha;} and {&beta;}. They lead to close interaction (positive feedback) and remote interaction (negative feedback), as a causative agent of the pattern formation on this model surfaces.<br>
-
[[File:IGKU0009.png|150px]][[File:IGKU0010.png|400px]]<br>
+
[[File:IGKU0009.png|150px]][[File:IGKU0011.png|300px]]<br>
-
Then let’s have a look on the interaction between two diffusible substances; one lead to close interaction (positive feedback) and other lead to remote interaction (negative feedback). Hereafter we name this diffusible substances A and B. A has large diffusion velocity and represses B’s increase. B has the small diffusion velocity and promotes both A and B’s increase. If A and B has this characteristic, close interaction (positive feedback) and remote interaction (negative feedback) are formed. A and B forms legato density gradient due to this interaction. When each cell units have the character “Color the appropriate color answering to the denser one among the two diffusible substances inside the cell unit,the substances’ density gradient can be imagined as pattern of cells.
+
Then let’s have a look on the interaction between two diffusible substances; one leads to close interaction (positive feedback) and other leads to remote interaction (negative feedback). Hereafter we name these diffusible substances A and B. A has large diffusion velocity and represses B’s increase. B has a small diffusion velocity and promotes both A and B’s increase. If A and B have this characteristic, close interaction (positive feedback) and remote interaction (negative feedback) are formed. <br>
 +
[[File:IGKU0010.png|500px]]<br>
 +
A and B forms legato density gradient due to this interaction. When each cell units have the character, “Follow the color of the diffusion substances with the higher density of the two”, the substances’ density gradient can be imagined by patterns of the cells.
<br>
<br>
-
Now let’s consider about how these two diffusible substances interacting each other in each cell unit. The amount of two diffusible substances in each cell unit changes only by diffusion and interaction. Then let’s focus on a certain cell unit (i) and consider about the concentration change. The substances amount change by diffusion is the difference between outflow and inflow. Change by the interaction is dependent on the amount of A and B at the certain moment.  
+
[[File:IGKU0050.png|400px]]<br>
 +
[[File:IGKU0030.png|400px]]<br>
 +
 
 +
Now let’s consider how these two diffusible substances interact with each other in each cell unit. The amount of two diffusible substances in each cell unit changes only by diffusion and interaction. Then let’s focus on a certain cell unit (i) and consider the change of the concentration. The change in the amount of substances by diffusion is caused by the difference between outflow and inflow. Change by the interaction is dependent on the amount of A and B at a certain moment.
<br>
<br>
-
[[File:Hannou.png]]<br>
+
[[File:Hannou.png|400px]]<br>
-
[[File:Hannou2.png]]
+
[[File:Hannou2.png|300px]]
<br>
<br>
-
Actually, this formula is the same as reaction-diffusion which is proposed by Turing for the purpose of explaining each factors of Turing pattern formation.
+
Actually, this formula is the same as reaction-diffusion which is proposed by Turing for the purpose of explaining each factors of Turing pattern formation. It seems to be difficult to understand the content of these formulae. We’re going to explain the content. <br>
-
It seems to be difficult to understand the content of these formulae. We’re going to explain the content.
+
[[File:ki1.png|18px]] , [[File:ki2.png]] and [[File:ki3.png]] are the constant numbers which indicates how big the influence on interaction of each diffusible substances per unit quantity is.
 +
In other words, these terms returns the amount of A and B at a moment if we substitute the amount of A and B at an anterior moment.
 +
[[File:dia.png]] and [[File:dib.png]] are the constant numbers peculiar to each diffusible substance which indicates the tendency of diffusion of A and B here. In other words, the terms [[File:dia.png]] and [[File:dib.png]] are the superficial inflow-outflow budget depending on diffusion of A and B. This is the contents which are described by the equitation.
 +
 
 +
<br>
 +
[[File:IGKU0012.png|300px]][[File:IGKU0013.png|200px]]<br>
 +
[[File:IGKU0014.png|300px]][[File:IGKU0015.png|300px]]<br>
<br>
<br>
-
Ki, Ki’, and Ki’’ are the constant numbers which indicates how big the influence on interaction of each diffusible substances per unit quantity is. In other words, these terms returns the amount of A and B at a moment depending on the interaction from the amount of A and B at anterior moment. DiA and DiB are the constant numbers peculiar to each diffusible substances which indicates the tendency of diffusion of A and B here. In other words, the terms DiA and DiB is the superficial inflow-outflow budget depending on diffusion of A and B. This is the contents which is described by the equitation.
 
-
<br><br>
 
==Experiments==
==Experiments==
-
We focused on the constants "Ki, Ki’, Ki’’" in these formulas. These are took as a given as "always fixed in any point" to Turing pattern. However, in fact, is it true that Ki is always fixed in any point with Turing pattern formed by E. coli? We thought it is not always true in wet work because E. coli makes A and B. In other words, increase or decrease speed of amount of A and B in a certain point depends on E. coli dencity in the point.<br>
+
We focused on the constants " [[File:ki1.png|18px]] , [[File:ki2.png]] , [[File:ki3.png]] " in these formulae. These are taken as "always fixed in any point" to Turing pattern. However, in fact, is it true that [[File:ki1.png|18px]] is always fixed in any point with Turing pattern formed by ''E. coli''? We thought it is not always true in wet work because ''E. coli'' makes A and B. In other words, increase or decrease speed of the amounts of A and B in a certain point depends on the ''E. coli'' density in the point.
-
As long as E. coli is growing not uniformly until a steady state, it must be generated E. coli density difference between each point. This E.coli density difference makes "Ki, Ki’, Ki’’" change between each point.<br>
+
 
-
Can we ignored "Ki, Ki’, Ki’’" difference? To confirm this, we established these assay.<br><br>
+
As long as ''E. coli'' is growing ununiformly until a steady state, ''E. coli'' density should be different between each point. This ''E. coli'' density difference makes changes of " [[File:ki1.png|18px]] , [[File:ki2.png]] , [[File:ki3.png]] " between each point.<br>
-
1. Confirm expression amount of GFP in both a steady state and a non-steady state with plated E. coli by common method.<br>
+
 
-
2. Confirm expression amount of GFP in E. coli that is activated other protein by IPTG and not activated E.coli as negative control and check whether expression amount of GFP depends on copy number<br>
+
Can we ignore the " [[File:ki1.png|18px]] , [[File:ki2.png]] , [[File:ki3.png]] " differences? To confirm this, we established these assays.<br><br>
 +
1. Confirm expression amount of GFP in both a steady state and a non-steady state with plated ''E. coli'' by common method.<br>
 +
[[File:IGKU0020.png|700px]]<br>
 +
2. how strong the other proteins' expression influences the expression of GFP. Make a comparison between a state with IPTG and with no IPTG.<br>
 +
[[File:IGKU0021.png|500px]]<br>
 +
 
 +
==Result==<br>
 +
[[File:GuchaFP1.png]]<br>
 +
[[File:GuchaFP2.png]]<br>
 +
We plated transformant ''E. coli'' containing GFP generator on 2 plates at same time.<br>
 +
It is evident that ''E. coli'' density is ununiformly. Moreover, the two plates shows completely different distribution.
==Discussion==
==Discussion==
-
Thus, when you plate E. coli by usual method, the E. coli express GFP ununiformly. This is because you can not plate E. coli enough uniformly. As long as the expression of GFP is so ununiform, even if you set maximum area of cell-unit which is necessary to generate pattern on a plate, the gaps of average mass of GFP expression between cell-units are large enough. When you set an enough small area to generate pattern, you should plate so uniformly that you can consider the gap of mass of GFP expression between cell-units are small enough. So it is necessary to refine the plating method. For that, wet lab should plate many times, dry lab should analyze the results every time, reach the minimum area of cell-unit which we can consider the gap of average mass of GFP expression small enough, and provide the dates for we lab. And wet lab refine the method. Thus, if wet lab and dry lab understand enough and go some way along each other, you can construct more accurate and more reliable method.<br>
+
Thus, when you plate E. coli by a usual method, the E. coli expresses GFP ununiformly. This is because you cannot plate E. coli enough uniformly. As long as the expression of GFP is ununiform, even if you set maximum area of cell-unit which is necessary to generate a pattern on a plate, the gaps of average mass of GFP expression between cell-units are large enough. When you set enough small area to generate pattern, you should plate uniformly so that you can consider whether the gap of mass of GFP expression between cell-units are small enough. So it is necessary to refine the plating method because the fact that the plates show a different distribution means that distribution depends on the method of plating. So, wet lab should plate many times.
-
As we have seen, there are factor means E. coli density we must consider when we think intercellular system. On the other hand, when we think a system inside the cell, the factor E. coli density is unrelated and do not have to consider. Therefore, after this, we think system inside the cell
+
And, dry lab should analyze the results every time, evaluate the minimum area of the cell unit which we can consider the gap of average mass of GFP expression small enough, and provide the dates for we lab. And wet lab refines the method. Thus, if wet lab and dry lab understand enough and go some way along each other, you can construct more accurate and more reliable method.
 +
As we have seen, there is E. coli density which we have to consider as factors when we think about the intercellular system. On the other hand, when we think a system inside the cell, the factor E. coli density is unrelated and do not have to be considered. Therefore, from now, we’d like to think about the system inside the cell.
==Conclusion==
==Conclusion==
-
As we showed the example ’Turing Pattern’, the results of wet lab and dry lab are often different because of their lack of understanding and appreciation each other. If both of them provide more information and closely discuss together, wet lab may be able to make an experimental system which imitates the system dry lab approximated to the real system. And wet lab provide quantified dates of a value which are necessary to formularize. If dry lab get these dates, they can create formulae which are well adapted to real system and run well simplified simulation. And if wet lab receive the anticipation date, they will be able to find more interesting results. When dry lab and wet lab join hands like this example ’Turing Pattern’, you can overthrow the future that some experiments should fail. Then, biology would evolve faster.
+
As we showed in the example ’Turing Pattern’, the results of wet lab and dry lab are often different because of their lack of understanding and appreciation of each other.  
 +
If both of them provide more information and closely discuss together, wet lab may be able to make an experimental system which imitates the system dry lab approximated to the real system.  
 +
And wet lab provides quantified data of a value which are necessary to formularize. If dry lab gets these data, they can create formulae which are well adapted to a real system, and run a well simplified simulation. And if wet lab receives the anticipation data, they will be able to find more interesting results. When dry lab and wet lab join hands like this example ’Turing Pattern’, you can overthrow the future that some experiments should fail. Then, biology would evolve faster.
 +
 
==Reference==
==Reference==
1:[http://www.sciencedirect.com/science/article/pii/S0092824005800084 A.M. Turing (1990) "The chemical basis of morphogenesis" Bulletin of Mathmatical Biology Vol. 52, No. 1/2, pp. 153-197]<br>
1:[http://www.sciencedirect.com/science/article/pii/S0092824005800084 A.M. Turing (1990) "The chemical basis of morphogenesis" Bulletin of Mathmatical Biology Vol. 52, No. 1/2, pp. 153-197]<br>

Latest revision as of 12:45, 10 October 2013

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Contents

Turing Model
-the problems between wet and dry-

Introduction

On Earth, there are various animals which have various patterns on their skin. The mechanism of this pattern formation has not been explained by any valid theories yet, although many hypothesis has been proposed. Among these hypothesis, there is a model pattern called Turing pattern proposed by A. Turing, a famous mathematician *1. S. Kondo *2 and some other researchers *3 suggest that some creatures’ pattern can be explained by Turing’s model. Here we will step by step explain how this Turing pattern is expressed by his model.
IGKU0002.pngSakana.png
Let’s take a look on a simple hypothetical pattern formed by just two colors. Creatures’ epidermal pattern is expressed on the cells. Let’s assume that the pattern is formed by cells in different state α and β for example. A cell in state α expresses color 1 and changes close cell in state β into state α. Another Cell in state β expresses color 2 and changes close cells in state α into state β, and remote cells in state β into state α. For convenience, hereafter we call the cell in state α {α}, and cell in the state β {β}.
IGKU0003.pngIGKU0004.png
Now, we will take a look at the system where two cells {α} and {β} exist uniformly and both of the cells are in the equilibrium state of interaction. Now suppose that the density of {α} and {β} fluctuated somewhere in the system. Assume that the density of cell {β} increases as shown in the center of figure 1. First, {β} in the center changes the neighboring {α} into {β}. And next the same {β} changes the remote {β} into {α}. Then remote {α} changes neighboring {β} into {α}. The pattern forms as this interaction continues
File:Stripeform.gif
Like this model, a striped pattern is formed by close-and-remote interactions between two states of cells. When we look at this close-and-remote interaction separately, close interaction can be explained as positive feedback reaction in the aspects of polarization. Conversely, the remote interaction can be explained by negative feedback reaction
IGKU0006.pngIGKU0001.png
Diffusing substances such as proteins secreted from the cells determine the characters of the cells. It seems that the characters which changes close or remote cells (i.e. α and β) are function of these diffusing substances. In other words, it can be said that {α} and {β} secretes different diffusing substances, and the substances lead to close interaction (positive feedback) and remote interaction (negative feedback). Therefore, the pattern formation can be said to be formed by interaction between diffusible substances, as well as cell-cell interaction.
IGKU0008.png
Then let’s consider about this interactions between diffusible substances in simplified model. Living organism’s body surface consists of cells shaped and sized ununiformly, therefore it is easier to understand if you assume that the body surface is a plane and consists of square cell-units sized uniformly. In this model, we can set diffusible substances which are secreted by {α} and {β}, and they increase and decrease under the influence of interactions. And then, these substances have the same characteristics of cell {α} and {β}. They lead to close interaction (positive feedback) and remote interaction (negative feedback), as a causative agent of the pattern formation on this model surfaces.
IGKU0009.pngIGKU0011.png
Then let’s have a look on the interaction between two diffusible substances; one leads to close interaction (positive feedback) and other leads to remote interaction (negative feedback). Hereafter we name these diffusible substances A and B. A has large diffusion velocity and represses B’s increase. B has a small diffusion velocity and promotes both A and B’s increase. If A and B have this characteristic, close interaction (positive feedback) and remote interaction (negative feedback) are formed.
IGKU0010.png
A and B forms legato density gradient due to this interaction. When each cell units have the character, “Follow the color of the diffusion substances with the higher density of the two”, the substances’ density gradient can be imagined by patterns of the cells.
IGKU0050.png
IGKU0030.png

Now let’s consider how these two diffusible substances interact with each other in each cell unit. The amount of two diffusible substances in each cell unit changes only by diffusion and interaction. Then let’s focus on a certain cell unit (i) and consider the change of the concentration. The change in the amount of substances by diffusion is caused by the difference between outflow and inflow. Change by the interaction is dependent on the amount of A and B at a certain moment.
Hannou.png
Hannou2.png
Actually, this formula is the same as reaction-diffusion which is proposed by Turing for the purpose of explaining each factors of Turing pattern formation. It seems to be difficult to understand the content of these formulae. We’re going to explain the content.
Ki1.png , Ki2.png and Ki3.png are the constant numbers which indicates how big the influence on interaction of each diffusible substances per unit quantity is. In other words, these terms returns the amount of A and B at a moment if we substitute the amount of A and B at an anterior moment. Dia.png and Dib.png are the constant numbers peculiar to each diffusible substance which indicates the tendency of diffusion of A and B here. In other words, the terms Dia.png and Dib.png are the superficial inflow-outflow budget depending on diffusion of A and B. This is the contents which are described by the equitation.


IGKU0012.pngIGKU0013.png
IGKU0014.pngIGKU0015.png

Experiments

We focused on the constants " Ki1.png , Ki2.png , Ki3.png " in these formulae. These are taken as "always fixed in any point" to Turing pattern. However, in fact, is it true that Ki1.png is always fixed in any point with Turing pattern formed by E. coli? We thought it is not always true in wet work because E. coli makes A and B. In other words, increase or decrease speed of the amounts of A and B in a certain point depends on the E. coli density in the point.

As long as E. coli is growing ununiformly until a steady state, E. coli density should be different between each point. This E. coli density difference makes changes of " Ki1.png , Ki2.png , Ki3.png " between each point.

Can we ignore the " Ki1.png , Ki2.png , Ki3.png " differences? To confirm this, we established these assays.

1. Confirm expression amount of GFP in both a steady state and a non-steady state with plated E. coli by common method.
IGKU0020.png
2. how strong the other proteins' expression influences the expression of GFP. Make a comparison between a state with IPTG and with no IPTG.
IGKU0021.png

==Result==
GuchaFP1.png
GuchaFP2.png
We plated transformant E. coli containing GFP generator on 2 plates at same time.
It is evident that E. coli density is ununiformly. Moreover, the two plates shows completely different distribution.

Discussion

Thus, when you plate E. coli by a usual method, the E. coli expresses GFP ununiformly. This is because you cannot plate E. coli enough uniformly. As long as the expression of GFP is ununiform, even if you set maximum area of cell-unit which is necessary to generate a pattern on a plate, the gaps of average mass of GFP expression between cell-units are large enough. When you set enough small area to generate pattern, you should plate uniformly so that you can consider whether the gap of mass of GFP expression between cell-units are small enough. So it is necessary to refine the plating method because the fact that the plates show a different distribution means that distribution depends on the method of plating. So, wet lab should plate many times. And, dry lab should analyze the results every time, evaluate the minimum area of the cell unit which we can consider the gap of average mass of GFP expression small enough, and provide the dates for we lab. And wet lab refines the method. Thus, if wet lab and dry lab understand enough and go some way along each other, you can construct more accurate and more reliable method. As we have seen, there is E. coli density which we have to consider as factors when we think about the intercellular system. On the other hand, when we think a system inside the cell, the factor E. coli density is unrelated and do not have to be considered. Therefore, from now, we’d like to think about the system inside the cell.

Conclusion

As we showed in the example ’Turing Pattern’, the results of wet lab and dry lab are often different because of their lack of understanding and appreciation of each other. If both of them provide more information and closely discuss together, wet lab may be able to make an experimental system which imitates the system dry lab approximated to the real system. And wet lab provides quantified data of a value which are necessary to formularize. If dry lab gets these data, they can create formulae which are well adapted to a real system, and run a well simplified simulation. And if wet lab receives the anticipation data, they will be able to find more interesting results. When dry lab and wet lab join hands like this example ’Turing Pattern’, you can overthrow the future that some experiments should fail. Then, biology would evolve faster.

Reference

1:[http://www.sciencedirect.com/science/article/pii/S0092824005800084 A.M. Turing (1990) "The chemical basis of morphogenesis" Bulletin of Mathmatical Biology Vol. 52, No. 1/2, pp. 153-197]
2:S. Kondo et al(2009) "How animals get their skin patterns: fish pigment pattern as a live Turing wave" Int. J. Dev. Biol. 53: 851-856
3:Akiko Nakamasu et al(2009) "Interactions between zebrafish pigment cells responsible for the generation of Turing patterns" PNAS vol. 106 no. 21 8429–8434