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 *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.
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 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.
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.
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.
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.


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.
「Ki, Ki’, Ki’’」は相互作用における各拡散物質の単位量あたりの影響の大きさを示す定数である。すなわち、ある瞬間におけるA,Bの量から相互作用のみによって増減した結果の次の瞬間のA,B量を返す。「DiA, DiB」はA,Bの拡散しやすさを表す各拡散物質固有の定数である。すなわち「○○、○○」はA,Bの拡散による見かけの流出入量である。 これが反応拡散方程式で表されている内容である。

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 (1) and (2) is the superficial inflow-outflow budget depending on diffusion of A and B. This is the contents which is described by the equitation.

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.
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 and check whether expression amount of GFP depends on copy number

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。
Thus, when you plate e-coli by usual method, the e-coli express GFP ununiformly. This is because you can’t 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.
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。
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

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:[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