Data from: Mutation rules and the evolution of sparseness and modularity in biological systems

Biological systems exhibit two structural features on many levels of organization: sparseness, in which only a small fraction of possible interactions between components actually occur; and modularity – the near decomposability of the system into modules with distinct functionality. Recent work suggests that modularity can evolve in a variety of circumstances, including goals that vary in time such that they share the same subgoals (modularly varying goals), or when connections are costly. Here, we studied the origin of modularity and sparseness focusing on the nature of the mutation process, rather than on connection cost or variations in the goal. We use simulations of evolution with different mutation rules. We found that commonly used sum-rule mutations, in which interactions are mutated by adding random numbers, do not lead to modularity or sparseness except for in special situations. In contrast, product-rule mutations in which interactions are mutated by multiplying by random numbers – a better model for the effects of biological mutations – led to sparseness naturally. When the goals of evolution are modular, in the sense that specific groups of inputs affect specific groups of outputs, product-rule mutations also lead to modular structure; sum-rule mutations do not. Product-rule mutations generate sparseness and modularity because they tend to reduce interactions, and to keep small interaction terms small.

生物系统在诸多组织层级上展现出两类结构特征：其一为稀疏性（sparseness），即组分间仅极小比例的潜在相互作用会实际发生；其二为模块化（modularity）——指系统可近乎分解为具备专属功能的模块的特性。既往研究表明，模块化可在多种情境下演化：比如随时间变化且共享相同子目标的目标集（模块化可变目标（modularly varying goals）），或是当连接存在成本时。本研究聚焦突变过程的本质（而非连接成本或目标变化），探讨模块化与稀疏性的起源，采用不同突变规则开展进化模拟实验。研究发现，常规使用的求和规则突变（sum-rule mutations，即通过添加随机数来改变相互作用强度的突变方式）仅在特殊情境下才能催生模块化与稀疏性；与之形成对比的是，乘积规则突变（product-rule mutations，即通过乘以随机数来改变相互作用强度——该模型更贴合生物突变的实际效应）可自然地产生稀疏性。当进化目标具备模块化属性（即特定输入组对应调控特定输出组）时，乘积规则突变同样可形成模块化结构，而求和规则突变则无法做到这一点。乘积规则突变之所以能产生稀疏性与模块化，是因为这类突变倾向于削弱相互作用，并维持微弱的相互作用项处于较低水平。