Team:Kyoto/ProjectTuring

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=Turing Model -the problems between wet and dry-=
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=Turing Model<br>-the problems between wet and dry-=
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==Motivation==
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Turing pattern is a theme which some teams of iGEM have already been working on. However, all of them have stopped at the dry step, and no team has actually succeeded in creating a pattern so far. What is the reason? This year, iGEM Kyoto considered the formation of Turing pattern, and attempted to span dry and wet approaches with a bridge.
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==Introduction==
==Introduction==
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Turing pattern is a kind of a mathematic model. Let’s imagine two diffusible substances which interact with each other. Among these two diffusible factors, one activates itself and the other, and the other represses itself.
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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>
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[[File:IGKU0002.png|300px]][[File:sakana.png|400px]]<br>
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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>
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[[File:IGKU0003.png|200px]][[File:IGKU0004.png|400px]]<br>
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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>
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[[File:stripeform.gif]]<br>
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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>
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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>
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[[File:IGKU0008.png|200px]]<br>
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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:IGKU0011.png|300px]]<br>
 +
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>
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[[File:IGKU0010.png|500px]]<br>
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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.
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<br>
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[[File:IGKU0050.png|400px]]<br>
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[[File:IGKU0030.png|400px]]<br>
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The two diffusible substances increase and decrease through interaction and create a difference in the density. This shading in the density causes the Turing pattern.
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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.
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<br>
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[[File:Hannou.png|400px]]<br>
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[[File:Hannou2.png|300px]]
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<br>
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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>
 +
[[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.
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Turing model is an abstract mathematic model. In order to understand intuitively, let me set a field divided into many cell units. In each cell there are the diffusible substances. Let’s consider how a pattern is formed here. Firstly, you focus on only one cell unit.
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<br>
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[[File:IGKU0012.png|300px]][[File:IGKU0013.png|200px]]<br>
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[[File:IGKU0014.png|300px]][[File:IGKU0015.png|300px]]<br>
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<br>
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Each of the two diffusible substances have two characters as we mentioned: “interaction affects to the amount of substances themselves” and “diffusibility”. This means that the density of the substances in the cell units changes moment by moment.
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==Experiments==
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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.
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Affecting the density, by in-flow and out-flow to the cell unit by a diffusion velocity peculiar to the substance (fig)
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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>
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Turing arranged these two factors and established the Reaction-diffusion equation below. (Hereafter we call the two diffusible substances A and B)
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Can we ignore the " [[File:ki1.png|18px]] , [[File:ki2.png]] , [[File:ki3.png]] " differences? To confirm this, we established these assays.<br><br>
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1. Confirm expression amount of GFP in both a steady state and a non-steady state with plated ''E. coli'' by common method.<br>
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[[File:IGKU0020.png|700px]]<br>
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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>
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[[File:IGKU0021.png|500px]]<br>
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<eq.>
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==Result==<br>
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[[File:GuchaFP1.png]]<br>
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[[File:GuchaFP2.png]]<br>
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We plated transformant ''E. coli'' containing GFP generator on 2 plates at same time.<br>
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It is evident that ''E. coli'' density is ununiformly. Moreover, the two plates shows completely different distribution.
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【日本語】
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==Discussion==
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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.
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「エニグマの解読などで知られる数学者A.Turing。彼の偉業の1つにTuring modelの考案がある。彼は数学者としての目線から、動物の体表にある模様の形成の謎を解こうとした。その結果彼が到達したのがTuring modelなのである。Turingいわく、動物の体表の模様もTuring patternに含まれるという。まだ完全に実証されたわけではないが、現に日本の近藤滋教授などによって、自然界におけるTuring patternの存在が再評価されつつある。」
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==Reference==
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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>
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Turing patternとはある種の数学モデルであり、「相互作用する2つの仮想因子」によってpatternを形成する。「相互作用する2つの仮想因子」とは拡散する因子であり、一方は自己と他方の増加を促進し、もう一方が他方の増加を抑制するというものだ。
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2:<html><a href="http://www.fbs.osaka-u.ac.jp/labs/skondo/paper/kondo%20IJDB%20review.pdf">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</a><br></html>
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()
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3:<html><a href="http://www.pnas.org/content/106/21/8429.short">Akiko Nakamasu et al(2009) "Interactions between zebrafish pigment cells responsible for the generation of Turing patterns" PNAS vol. 106 no. 21 8429–8434</a><br></html>
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このような性質を持つ2つの仮想因子が相互作用しあいながら増減し、その結果として因子の濃度にムラが生まれ、仮想因子の密度の濃淡によってpatternが浮かびあがるというのがTuring patternの略図である。
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</div>
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(図)
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Turing modelはあくまでも数学モデルである。これを直感的に把握するため、実際にある空間を仮定してpatternがいかにして形成されるかを考えてみよう。まずは解りやすくするために、空間を下図のようにいくつものcell unitに分割し、1つのcell unitについてのみ考えよう。
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(図)
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2つの仮想因子の濃度のムラはどのようにして形成されるだろうか。仮想因子の性質を思い出してみると「拡散し」「相互作用によって増減する」とある。つまり「相互作用による増減」と「拡散による流出入」によって仮想因子の濃度が刻一刻と変化するのだ。「相互作用による増減」はそれぞれの因子に固有の反応速度で、促進・抑制しあう。
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(図)
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そして「拡散による流出入」もそれぞれの因子に固有の拡散速度でcell unitの中に流出入しながらその濃度に影響を与える。
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(図)
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この2つの要素を、Turingは下に表す「反応拡散方程式」にまとめた。(便宜上2つの仮想因子をそれぞれA,Bと呼ぶ事とする)
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(反応拡散方程式)
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この2式を少し見ただけでは解りづらいと思うので、簡単な説明を加える。先ほど述べたように反応拡散方程式は、patternが形成される空間の、ある一定領域(i)における2つの仮想因子の増減を表している。
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(図)
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Ki, Ki’, Ki’’は領域iにおいてそれぞれ仮想因子が相互作用する際の増減変化の定数であり、これらは全て任意の領域iにおいて常に一定である。DiA, DiBはそれぞれの仮想因子に固有の「拡散しやすさ」を表す定数である。すなわち、反応拡散方程式はある時間の微小領域における2つの仮想因子A, Bの存在量に応じて、その両方が微小時間後にどう増減するかを表しているのだ。
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()
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私たちが今回目をつけたのは、この両式の中での「Ki, Ki’, Ki’’」という定数である。これらはTuring pattern形成の為に「任意の領域iにおいて常に一定である」という前提がある。
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()
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しかし大腸菌でTuring patternを形成する際には、これらは本当に任意のiにおいて常に一定だろうか。ここで私たちはwetにおいては成り立たない場合が生じてくると考えた。というのも、このTuring patternを大腸菌で実現しようとした時、A,Bが増加するのは大腸菌の合成によってであるため、「一定領域内の大腸菌密度」に「A,Bの増減速度」が依存してしまうからである。
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大腸菌が定常状態に至るまではまばらに生育している限り、どうしても大腸菌密度に差が生じてしまう。この密度差によって、「Ki, Ki’, Ki’’」がiによって異なると考えられるのだ。
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(図)
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それではこの「Ki, Ki’, Ki’’」の差は無視出来るものだろうか。これを確認するための実験系を考えた。dryとwetとの差を埋めるために、私たちはこの実験をdryと本実験との間に行うことを提案する。
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(図)
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==idea==
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==Discussion==
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==conclusion==
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  </div>
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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