Combinatorial explosion in model gene networks
Combinatorial explosion in model gene networks
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作者：
Edwards, R;Glass, L
人类和其他生物体基因组知识的爆炸性增长留下了一个问题，即如何协调相互作用网络中基因的功能以进行有序活动。解决这个问题的一种方法是研究抽象网络模型的数学性质，这些模型捕捉了基因网络的逻辑结构。主要的问题是了解特定的网络结构如何导致特定的活动模式，以及可能的行为类型。我们研究理想化模型，其中网络的逻辑结构由布尔函数明确表示，布尔函数可以由n-立方体上的有向图表示，但在时间上是连续的，并由微分方程描述，而不是通过离散时钟同步更新。该方程是分段线性的，这允许重要的分析，并有利于快速积分沿着轨迹。我们首先给出了一个组合的解决方案的问题，有多少不同的逻辑结构存在的n维网络，显示的数量增加非常迅速的n。然后，我们概述的分析方法，可用于建立的存在性，稳定性和周期的周期轨道对应于特定的周期的n-立方体。我们使用这些方法来确认极限环的存在，发现在一个样本的100万随机生成的网络结构的4个基因。即使只有4个基因，至少有数百种不同的稳定周期性行为模式是可能的，其中许多令人惊讶的复杂。我们讨论了进一步分类这些周期性行为的方法，表明小突变（反转的一个或几个边的n-立方体）不需要破坏极限环的稳定性。虽然这些网络作为基因网络的模型非常简单，但它们的数学透明度揭示了结构和行为之间的关系，它们表明在这种网络中有序动态的可能性非常丰富，并且它们提供了思考突变如何改变动态的新方法。(C)2000年美国物理学会。[S1054-1500（00）01103-4]。
The explosive growth in knowledge of the genome of humans and other organisms leaves open the question of how the functioning of genes in interacting networks is coordinated for orderly activity. One approach to this problem is to study mathematical properties of abstract network models that capture the logical structures of gene networks. The principal issue is to understand how particular patterns of activity can result from particular network structures, and what types of behavior are possible. We study idealized models in which the logical structure of the network is explicitly represented by Boolean functions that can be represented by directed graphs on n-cubes, but which are continuous in time and described by differential equations, rather than being updated synchronously via a discrete clock. The equations are piecewise linear, which allows significant analysis and facilitates rapid integration along trajectories. We first give a combinatorial solution to the question of how many distinct logical structures exist for n-dimensional networks, showing that the number increases very rapidly with n. We then outline analytic methods that can be used to establish the existence, stability and periods of periodic orbits corresponding to particular cycles on the n-cube. We use these methods to confirm the existence of limit cycles discovered in a sample of a million randomly generated structures of networks of 4 genes. Even with only 4 genes, at least several hundred different patterns of stable periodic behavior are possible, many of them surprisingly complex. We discuss ways of further classifying these periodic behaviors, showing that small mutations (reversal of one or a few edges on the n-cube) need not destroy the stability of a limit cycle. Although these networks are very simple as models of gene networks, their mathematical transparency reveals relationships between structure and behavior, they suggest that the possibilities for orderly dynamics in such networks are extremely rich and they offer novel ways to think about how mutations can alter dynamics. (C) 2000 American Institute of Physics. [S1054-1500(00)01103-4].