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数学在生物中的应用和滥用

已有 7063 次阅读 2012-9-8 00:26 |个人分类:工作笔记|系统分类:科研笔记| 数学, 生物, 模型, 计算, 定量

前两天有博主讨论数学公式和引用率的问题。我今天偶然看到一篇题目叫做“数学在生物中的应用和滥用”的发在Science上的2004年的老文章(英文原名 "Uses and Abuses of Mathematics in Biology",原文PDF在博文最下方下载)。觉得里面的东西在今天读来还是有些意义。

(本来想偷懒,少写点儿东西,贴一个PDF就好了。但是编辑把这个博文放到主页上了,我觉得还是把这篇文章大概总结一下,才对得起各位读者)

主要内容点:
1. 数学在生物学发展的历史上起到过重大的作用。譬如Hardy and Weinberg equilibrium 对种群遗传的公式描述和Fisher, Haldane, and Wright等人在20世纪初期对于生物数学做出了进一步的发展。这里顺便说一句,Fisher(费希尔)这个名字可能对于大多数人来说是作为一个著名的统计学家出现的。他其实还是一个生物遗传学家。
2.在一般的教学中,往往是先教理论性,然后教应用性的。而现在的新新人类可能会先有应用性的经验,譬如在电脑游戏里不经意用到的统计学。
3.数理科学注重于抽象性,而生物系统较为复杂。最近的计算工具改革促进了生物研究的计算模型化。
4.作者把数学和计算在生物中的兴起比喻成“牛顿力学”的初级阶段
5.计算工具的进步使得更多的生物学家们能够利用数学模型做研究,但是很多人缺少数学基础。他们不一定明白自己在搞啥。
6.复杂的模型不一定是更有意义的,更接近现实的,更有预见力的。对于不知道的参数做出假设是危险的。作者举了世界卫生组织和纽约人口协会做的一个研究艾滋病在非洲传染机率的例子。这个研究应用了复杂的公式,但其中的一个假设是没有意义的:一个人和10个不同的人性交后被传染机率跟和同一个人性交10次的机率是相同的。很显然,前者的机率应该是更高的。第二个例子是英国做的一个关于食用带骨牛肉对于克雅氏病传染率的研究。研究者运用了数学模型,其结论是如果禁止食用带骨牛肉,就最少可以避免一例感染。结果,英国政府就颁布了这个禁令。而这个研究里面对于一个参数 - 食用了一定量感染病毒的牛肉后会被传染的机率 - 做了没有根据的泛泛的假设。但是在当时是不知道这个参数到底是多少的。本文的作者用了另外一种办法做了估计,结论是禁止食用带骨牛肉只会减少一个传染例子的一千分之一[也就是把一个人劈成一千份儿]。事实证明,英国的群众的眼睛是雪亮的,大家都觉得这个禁令是无理取闹,极其可笑的。
7.历史上数学的用处在不同的生物学科中是不一样的。生态学对于数学的应用较晚,到20世纪六十年代才开始。之前大多数是描述性的。很多野外作业的生态学家对于模型最开始是嗤之以鼻的,但是在今日,生态学是一个又有野外考察、实验观察,又有数学分析的学科。相比较,免疫学应用数学就更晚了,但是分子水平的计算和模型对于设计药物有很大的帮助。
8.数学和生物的结合相对于物理和工程才刚刚开始。有些‘滥用’是难免的。最大的问题是用了一大堆复杂的参数和细节,却对于一些关键性的参数做出猜测。想想爱因斯坦说的一句话吧:尽管不要过度简单,但是模型还是越简洁越好[我自己的翻译。我不太同意这句话在网上的翻译]。

一些有意思的原文摘录:
“The virtue of mathematics in such a context is that it forces clarity and precision upon the conjecture, thus enabling meaningful comparison between the consequences of basic assumptions and the empirical facts.”
“A point that arguably deserves more emphasis than it usually gets is that, in such exploration of mathematical models [of biological/evolutionary questions], the understanding emerging from complex computer-based simulations can often be substantially less complete than that from the analytic methods of classical applied mathematics and theoretical physics.”
"...an increasingly large body of work ... are drawn from the alleged working of a mathematical model, without clear understanding of what is actually going on"
“Sadly, examples of the application of statistical “confidence intervals” to distributions resulting from making arbitrary assumptions about essentially unknown parameters, and then endowing this with reality by passage through a computer, continue to proliferate.”
“...there is as yet no agreed explanation for why there is so long, and so variable, an interval between infection with HIV and onset of AIDS...It may even be that the design of effective vaccines against protean agents like HIV or malaria will require such population-level understanding."
"More familiar in some areas than others are the benefits of mathematical studies that underpin pattern seeking and other software that is indispensable in elucidating genomes, and ultimately in understanding how living things assemble themselves. Very generally useful are still-unfolding advances that illuminate the frequently counterintuitive behavior of nonlinear dynamical systems of many kinds."
"Mathematics, however, does not have the long-standing relation to the life sciences that it does to the physical sciences and engineering."
"Particularly tricky are instances in which conventional statistical packages (often based on assumptions of an underlying Gaussian distribution—the central limit theorem) are applied to situations involving highly nonlinear dynamical processes (which can often lead to situations in which 'rare events' are significantly more common than Gaussian distributions suggest)"
"Perhaps most common among abuses, and not always easy to recognize, are situations where mathematical models are constructed with an excruciating abundance of detail in some aspects, whilst other important facets of the problem are misty or a vital parameter is uncertain to within, at best, an order of magnitude. It makes no sense to convey a beguiling sense of “reality” with irrelevant detail, when other equally important factors can only be guessed at. Above all, remember Einstein’s dictum:'models should be as simple as possible,but not more so.'"

文章摘要:
In the physical sciences, mathematical theory and experimental investigation
have always marched together. Mathematics has been less intrusive in the life
sciences, possibly because they have until recently been largely descriptive,
lacking the invariance principles and fundamental natural constants of physics.
Increasingly in recent decades, however, mathematics has become pervasive in
biology, taking many different forms: statistics in experimental design; pattern
seeking in bioinformatics; models in evolution, ecology, and epidemiology; and
much else. I offer an opinionated overview of such uses—and abuses.


全文请下载PDF:

May 2004 Uses and abuses of mathematics in biology.pdf



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