数理生物学 1952

形态发生的化学基础

艾伦·图灵

两种化学物质，一边反应、一边扩散，就能把均匀的组织变出斑点与条纹。

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In depth · the introduction

在一个胚胎长出任何斑点、条纹或四肢之前，它不过是一团几乎毫无特征、彼此一模一样的细胞。这篇论文解释了：它如何能给自己铺出图案——而且全无蓝图。

把这个想法拆开看

生命里满是规整的图案：豹斑那均匀的间距、斑马的条纹、苍蝇身上的刚毛、绕着茎排开的芽。若细胞起初都一样，这间距又从何而来？图灵的回答是：图案，能用化学把自己造出来。

设想两种化学物质在组织里飘移，一边走一边反应。一种会促进自己，却只渗出很短一段距离；另一种——也由前者造出——铺得远得多，并把前者压下去。真正出人意料的，是它们合在一起所做的事。扩散，平时把差异抹成一片均匀的模糊，在这里却反其道而行——它把那些细小的、偶然的隆起，磨利成一种规整、重复的图案。图灵证明了这是可能的，并称之为「扩散驱动失稳」。

它从哪里来

艾伦·图灵于 1952 年在曼彻斯特写下此文，那时距他战时的破译工作、以及他为计算机奠基的工作，已过了几年。他从机器转向生命的形态，用世上最早的电子计算机之一——Ferranti Mark I——算出了一幅斑驳的图案，这是整个生物学里最早的计算机模拟之一。它也是他最后的作品之一。1952 年，他因身为同性恋者被起诉，被迫接受激素治疗，并于 1954 年去世，这项工作未及完成。

它为何重要

它让生物学能「免费」获得结构。你不需要一位小小的建筑师去逐个安放每一个斑点；一条简单的化学规则，在所有地方同时、一遍遍地重复，就自己把整幅图案铺了下来。这个想法——秩序能从均匀的开端中涌现，一个系统能自我组织——成了我们理解发育的一块基石，而它所及，远不止动物的皮毛。

一种想象的方式

想想人群里的一则谣言。一则有料的谣言会自我繁殖，传给紧挨着你的人——这是一个短程的放大器。可它同时会触发一句传得更快的辟谣，抢先一步跑到前面，把更远处的谣言压下去——这是一个长程的阻尼器。从一群平静、均匀的人开始，你最后得到的，不是一片均匀的低语，而是一处处相信谣言的人群，彼此间距均匀地隔开。把「谣言」换成「色素」，你就有了一头豹子的雏形。

它落在何处

图灵在本馆出现了三次，每一次都站在一个起点上。他定义了「计算」本身（1936），并追问机器能否思考（1950）；在这里，他追问一具身体如何造出自己的形状。达尔文（1859）解释了形态为何会跨代改变，孟德尔（1866）解释了性状如何传递，而图灵追问的是：一个受精卵，如何在物理上长成一个有图案的生物。他那些反应着、铺展着的化学物质，回响着洛特卡与沃尔泰拉（1926）的捕食者–猎物循环——同样是「彼此制造、彼此消耗」之物的数学。

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