Turing Model
-the problems between wet and dry-

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 [citation needed*2] and some other researchers [citation needed*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.
Let’s take a look on simple hypothetical pattern formed by just two colors. Creatures’ epidermal pattern is expressed on the cell. Let’s assume that the pattern is formed by cells in different state α and β for example. Cell in state α expresses color 1 and changes close cell in state β into state α. Cell in state β expresses color 2 and change close cell in state α into state β, and remote cell in state β into cell in state α. For convenience, hereafter we call the cell in the state of α by {α} and cell in the state of β by {β}. Then, we will take a look on the system that two cells {α} and {β} are existing uniformly and both of the cells are in the equilibrium state of interaction. Now suppose that the density of {α} and {β} fluctuated in somewhere in the system. Assume that the density of cell {β} increases like the cells in center of figure 1. At first, the center {β} changes close {α} into {β}. And next same {β} changes remote {β} into {α}. Then remote {α} changes close {β} into {α}. The pattern is formed as this interaction continues.
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.
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 {&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.
それではこの構図をより単純化して考えてみよう。生物の体表は形や大きさの不均一な細胞によって構成されているため、体表を均一な大きさの正方形cell unitsが並んだ平面として単純化して考えた方が解りやすい。また、この平面上でせめぎ合う拡散物質（前の段落で述べてた、具体的にどういうものか？）が「相互作用し合う事で増減する」性質をもつ2種類の拡散物質であるとしてもよい。この時、2種類の拡散物質が先ほどの細胞で出てきたような「近距離におけるポジティブフィードバック」と「遠距離におけるネガティブフィードバック」を生じさせる物質だとする。
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, they are substances which has the same characteristic of cell {α} and {β} (substances lead to close interaction (positive feedback) and remote interaction (negative feedback)), as a causative agent of the pattern formation on this model surfaces.

それでは、cell units上で拡散物質が「近距離におけるポジティブフィードバック」と「遠距離におけるネガティブフィードバック」を生じるような相互作用について考えよう。この拡散物質の名前をA,Bとすると、 下図に示すような性質を拡散物質A,Bが持つとする。すなわち、「Aは拡散速度が大きくBの増加を抑制する」性質を持ち「Bは拡散速度が小さくA,B両方の増加を促進する」性質を持つとする。これを、近距離・遠距離で分けて考えると、下図のように近距離ではポジティブフィードバックを、遠距離ではネガティブフィードバックを形成するのである。 この相互作用によって、AとBは滑らかな濃度勾配を持つ分布を形成する。各Cell unitは「その内部にある2種類の拡散物質の濃度の高い方に応じた呈色をする」性質を持つとすると、拡散物質の量が可視化されるので、これまでの議論より拡散物質が形成する分布に応じてcell unitsがpatternを形成するはずだ
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.

Experiments

We focused on the constants "Ki, Ki’, Ki’’" in these formula. 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.
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.
Can we ignored "Ki, Ki’, Ki’’" difference? To confirm this, we established these assay.

1. Confirm expression amount of GFP in both a steady state and a non-steady state with plated E. coli by common method.
2. Confirm expression amount of GFP in E. coli that is activated other protein by IPTG and not activated E.coli as negative control
3. Confirm if expression amount of GFP depends on copy number with construction in Assay2.

Discussion

このように、通常のメソッドで大腸菌をまくと、GFPの発現量が非常にまばらになる。これは大腸菌をまく際に十分均一にまく事ができていないからである。これほどまばらな発現量の状態だと、シャーレ上に模様を形成するために必要な最大面積のcell unitを仮定したときであってもcell unit内での平均GFP発現量の差異が誤差の範囲とならない。Patternを形成するために十分小さなcell unitを仮定した時に、cell unit内の平均GFP発現量が誤差として扱える程度まで均一にまかなければならないため、ここから更にメソッドを洗練させていく必要がある。その為にも、wetが繰り返し大腸菌をまく作業を行い、dryがその結果を毎回確認して、平均GFP発現量が誤差の範囲となる最小のcell unit面積を求め、そのデータをwetに還元し、wetがメソッドの精度をあげていかなければならない。このようにして、wetとdryとが互いを十分理解し、歩み寄ることによってより正確で信頼できるメソッドの構築ができる。
見てきたとおり、細胞間での回路を考えると、考えなければならないファクターとして菌体密度という要素が増える。細胞内での回路を考えた場合、考えているものは一細胞内なので、菌体密度の要素を考えずに済む。よって、その要素のない回路を以降考えてみたい。

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.

Reference

1:A.M. Turing (1990) "The chemical basis of morphogenesis" Bulletin of Mathmatical Biology Vol. 52, No. 1/2, pp. 153-197
2:S. Kondo(2009) "How animals get their skin patterns: fish pigment pattern as a live Turing wave" Int. J. Dev. Biol. 53: 851-856
3: