一类生物网络多稳态参数发生的确定
Identifying the Parametric Occurrence of Multiple Steady States for some Biological Networks
摘要：
我们考虑一个问题，从生物网络分析确定区域的参数空间中，有多个稳定状态的变量和参数的正实值。我们描述了使用符号计算中的工具来解决这个问题的多种方法。我们描述了如何取得进展，实现半代数描述的参数空间的多平稳区域，并比较符号结果的数值方法。所研究的生物网络是丝裂原活化蛋白激酶（MAPK）网络的模型，该网络已经消耗了相当大的努力，使用相应模型的结构的特殊见解。我们的主要例子是一个包含11个变量和19个参数的11个方程的模型，其中3个是符号处理的兴趣。该模型还对所有变量和参数施加了积极条件。我们应用组合的符号计算方法设计的混合等式/不等式系统，特别是虚拟替代，懒惰的真正的三角化和柱形代数分解，以及适应高斯消除和图论的简化技术。我们能够确定我们的主要例子在2维参数空间的多平稳性。我们还研究了第二个MAPK模型和一个符号网格采样技术，可以定位这样的区域在三维参数空间。
英文摘要：
We consider a problem from biological network analysis of determining regions in a parameter space over which there are multiple steady states for positive real values of variables and parameters. We describe multiple approaches to address the problem using tools from Symbolic Computation. We describe how progress was made to achieve semi-algebraic descriptions of the multistationarity regions of parameter space, and compare symbolic results to numerical methods. The biological networks studied are models of the mitogen-activated protein kinases (MAPK) network which has already consumed considerable effort using special insights into its structure of corresponding models. Our main example is a model with 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment. The model also imposes positivity conditions on all variables and parameters. We apply combinations of symbolic computation methods designed for mixed equality/inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition, as well as a simplification technique adapted from Gaussian elimination and graph theory. We are able to determine multistationarity of our main example over a 2-dimensional parameter space. We also study a second MAPK model and a symbolic grid sampling technique which can locate such regions in 3-dimensional parameter space.
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