數理生物學 1952

形態發生的化學基礎

艾倫·圖靈

兩種化學物質，一邊反應、一邊擴散，就能把均勻的組織變出斑點與條紋。

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In depth · the introduction

在一個胚胎長出任何斑點、條紋或四肢之前，它不過是一團幾乎毫無特徵、彼此一模一樣的細胞。這篇論文解釋了：它如何能給自己鋪出圖案——而且全無藍圖。

把這個想法拆開看

生命裡滿是規整的圖案：豹斑那均勻的間距、斑馬的條紋、蒼蠅身上的剛毛、繞著莖排開的芽。若細胞起初都一樣，這間距又從何而來？圖靈的回答是：圖案，能用化學把自己造出來。

設想兩種化學物質在組織裡飄移，一邊走一邊反應。一種會促進自己，卻只滲出很短一段距離；另一種——也由前者造出——鋪得遠得多，並把前者壓下去。真正出人意料的，是牠們合在一起所做的事。擴散，平時把差異抹成一片均勻的模糊，在這裡卻反其道而行——它把那些細小的、偶然的隆起，磨利成一種規整、重複的圖案。圖靈證明了這是可能的，並稱之為「擴散驅動失穩」。

它從哪裡來

艾倫·圖靈於 1952 年在曼徹斯特寫下此文，那時距他戰時的破譯工作、以及他為電腦奠基的工作，已過了幾年。他從機器轉向生命的形態，用世上最早的電子電腦之一——Ferranti Mark I——算出了一幅斑駁的圖案，這是整個生物學裡最早的電腦模擬之一。它也是他最後的作品之一。1952 年，他因身為同性戀者被起訴，被迫接受激素治療，並於 1954 年去世，這項工作未及完成。

它為何重要

它讓生物學能「免費」獲得結構。你不需要一位小小的建築師去逐個安放每一個斑點；一條簡單的化學規則，在所有地方同時、一遍遍地重複，就自己把整幅圖案鋪了下來。這個想法——秩序能從均勻的開端中湧現，一個系統能自我組織——成了我們理解發育的一塊基石，而它所及，遠不止動物的皮毛。

一種想像的方式

想想人群裡的一則謠言。一則有料的謠言會自我繁殖，傳給緊挨著你的人——這是一個短程的放大器。可它同時會觸發一句傳得更快的闢謠，搶先一步跑到前面，把更遠處的謠言壓下去——這是一個長程的阻尼器。從一群平靜、均勻的人開始，你最後得到的，不是一片均勻的低語，而是一處處相信謠言的人群，彼此間距均勻地隔開。把「謠言」換成「色素」，你就有了一頭豹子的雛形。

它落在何處

圖靈在本館出現了三次，每一次都站在一個起點上。他定義了「計算」本身（1936），並追問機器能否思考（1950）；在這裡，他追問一具身體如何造出自己的形狀。達爾文（1859）解釋了形態為何會跨代改變，孟德爾（1866）解釋了性狀如何傳遞，而圖靈追問的是：一個受精卵，如何在物理上長成一個有圖案的生物。他那些反應著、鋪展著的化學物質，迴響著洛特卡與沃爾泰拉（1926）的捕食者–獵物循環——同樣是「彼此製造、彼此消耗」之物的數學。

The original document

Original source text

A. M. Turing · Philosophical Transactions of the Royal Society of London B, vol. 237, no. 641, pp. 37–72 · 14 August 1952

Abstract

It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis.

Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances.

A model of the embryo · morphogens

These substances will be called morphogens, the word being intended to convey the idea of a form producer.

This model will be a simplification and an idealization, and consequently a falsification.

Turing opens by stating his aim and his method: not to model any one organism in detail, but to ask whether the simplest possible chemistry — substances that react and diffuse — can in principle generate biological form. He is candid that the picture is idealised, and treats the cell as a vessel of reacting morphogens whose concentrations vary in space and time.

[ … ]

The breakdown of symmetry and homogeneity

The heart of the paper is an argument that the spatially uniform state can be unstable. Turing linearises the reaction–diffusion equations about a homogeneous equilibrium and follows how a small disturbance grows. Crucially, diffusion — ordinarily a force that smooths differences away — is shown to be capable of doing the opposite when two morphogens diffuse at different rates: it can amplify disturbances of a particular size.

Reaction and diffusion on a ring of cells

To make the analysis concrete he studies a ring of discrete cells and, equivalently, a continuous medium, decomposing any disturbance into its spatial waves. Each wavelength grows or decays on its own; a band of them grows, and the fastest-growing wave fixes the spacing of the emerging pattern. Turing catalogues six qualitatively distinct outcomes, from a return to uniformity, through stationary waves of definite wavelength — the spots-and-stripes case — to oscillations both standing and travelling.

A computed dappled pattern

Going beyond the linear analysis, Turing integrates the full nonlinear equations numerically on the Ferranti Mark I computer at Manchester to produce a two-dimensional ‘dappled’ pattern — among the earliest computer simulations of a biological process. He also sketches how the same instability might govern gastrulation and the spiral, Fibonacci-numbered arrangement of leaves (phyllotaxis).

[ … ]

The full thirty-six-page memoir — with its linearised equations, its catalogue of the six behaviours, the ring-of-cells calculation, and the computed dappled figure — is available in full at the source below.

Manchester · 1952