The transition and properties from Boolean networks to discrete-time Markov chains: A case study of mice stem cell gene regulatory networks
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摘要: 提出了一种基于概率模型检测技术的新方法用于解决生物工程中基因调控网络探查吸引子这一关键问题.以大鼠干细胞基因调控网络的吸引子找寻这样一个具体问题为例，将布尔网络表示的基因调控网络的更新函数通过对应的真值表，转换为离散时间马尔科夫链，写入模型检测工具PRISM中；之后通过验证模型的系统性质的技术去验证每个基因在很长一段时间之后的激活概率，以此找到基因调控网络中的吸引子.同时，通过添加基因扰动的方式，改变每个基因的激活/抑制概率，可以找到每个基因对其他基因的促进/抑制关系.实验表明，大鼠干细胞基因中有7个基因在一段时间后状态不变，剩余基因的变化共同构成了一个吸引环.整个检测流程简洁易用，可以直接找出吸引子.进一步地，实验准确地找出了大鼠干细胞中Gata1基因的抑制/促进对象，此实验结果对解决大鼠的白血球减少症有着治疗方面的意义.Abstract: This paper proposes a new method based on probabilistic model checking technique to solve the problem of detecting attractors in gene regulatory networks which is vital in bio-engineering. We transform a gene regulatory network into a discrete-time Markov chains (DTMC for short) by using truth table. We verify the possibility of gene activation in some "long term" through a model checker named PRISM, which help us to find attractors of the gene regulatory network. In this paper, we show the whole procedure using the example of mice stem cell gene regulatory networks. Meanwhile, we make a new technique of detecting the promotion/inhibition relations between genes by adding gene disturbance and modifying gene activation/suppression probability. We show that in the mice regulatory network, seven genes will remain their invariable states in some "long term", then the rest of "changing" genes forms a cyclic attractor. The experiment shows that our method can find the attractors easily and directly. Moreover, our experiment successfully finds those genes affected by gene Gata1, which would be helpful for studying the mice leucopenia.
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Key words:
- Boolean networks /
- discrete-time Markov chains /
- gene regulatory network
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图 1 a) 布尔网络单吸引子举例; (b)布尔网络吸引环举例
Fig. 1 (a) An example of Boolean network singleton attractors; (b) An example of Boolean network cyclic attractors
图 3 布尔网络转换为DTMC模型技术路线
Fig. 3 The transition technical route from Boolean network to DTMC
表 1 EKLF的更新函数转换产生的真值表
Tab. 1 The truth table of gene EKLF's update function
EKLF Gata1 Fli1 0 0 0 0 0 1 1 1 0 0 1 1 
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表 2 大鼠干细胞基因布尔更新函数
Tab. 2 Mice stem cell gene regulatory network's Boolean update function
Gene Update function Gata 2 $\rm Gata2\wedge\neg(Pu.1\vee(Gata1\wedge Fog1))$ Gata1 $\rm (Gata1\vee Gata2\vee Fil1)\wedge\neg Pu.1$ Fog1 Gata1 EKLF $\rm Gata1\wedge\neg Fli1$ Flil $\rm Gata1\wedge\neg EKLF$ Scl $\rm Gata1\wedge\neg Pu.1$ Cebpa $\rm Cebpa\wedge\neg(Scl\vee(Fog1\wedge Gata1))$ Pu.1 $\rm (Cebpa\vee Pu.1)\wedge\neg(Gata1\vee Gata2)$ cJun $\rm Pu.1\wedge\neg Gfi1$ EgrNab $\rm (Pu.1\wedge cJun )\wedge\neg Gfi1$ Gfi1 $\rm Cebpa\wedge\neg EgrNab$
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表 3 吸引子找寻相关信息计算结果
Tab. 3 The result of attractor finding information
公式 基因长期激活概率 基因种类 $S$=?[Gata2=1] 0 抑制基因 $S$=?[Gata1=1] 0 $S$=?[Fog1=1] 0 $S$=?[EKLF=1] 0 $S$=?[Fli1=1] 0 $S$=?[Scl=1] 0 $S$=?[Cebpa=1] 0.999 99 激活基因 $S$=?[Pu.1=1] 0.999 99 $S$=?[cJun=1] 0.25 变化基因 $S$=?[EgrNab=1] 0.25 $S$=?[Gfi1=1] 0.75 公式 基因在特定时刻激活概率 基因种类 $T=1$ $T=2$ $T=100$ $P$=?$[F[T, T]$Gata2=1] 0.666 6 0.222 2 3.72E-44 抑制基因 $R$=?$[F[T, T]$Gata1=1] 0 0 0 $R$=?$[F[T, T]$Fog1=1] 0 0 0 $R$=?$[F[T, T]$EKLF=1] 0 0 0 $R$=?$[F[T, T]$Fli1=1] 0 0 0 $R$=?$[F[T, T]$Scl=1] 0 0 0 $R$=?$[F[T, T]$Cebpa=1] 1 1 1 激活基因 $R$=?$[F[T, T]$Pu.1=1] 1 1 1
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表 4 基因Gata1的过度表达扰动结果
Tab. 4 The result of gene Gata1's over expression
公式 基因长期激活概率 无基因扰动结果 加入Gata1过度表达扰动结果 $S$=?[Gebpa=1] 0.999 99 0.876 $S$=?[cJun=1] 0.25 0.342 9 $S$=?[EgrNab=1] 0.25 0.342 9 $S$=?[Gfi1=1] 0.749 9 0.657 公式 基因在某时刻前激活时间统计$T=100$ 无基因扰动结果 加入Gata1过度表达扰动结果 $R${"Gata1active"}=?[$C<=T$] 0 2.098 $R${"Fog1active"}=?[$C<=T$] 0 1.12 $R${"EKLFactive"}=?$[C<=T]$ 0 1.343 $R${"Gebpaactive"}=?$[C<=T]$ 100 88.387 $R${"cJunactive"}=?$[C<=T]$ 25.277 7 33.693 $R${"EgrNabactive"}=?$[C<=T]$ 24.361 1 32.394 $R${"Gfi1active"}=?$[C<=T]$ 73.694 4 65.034
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