论文标题
输入输出网络中无限稳态的结构
The Structure of Infinitesimal Homeostasis in Input-Output Networks
论文作者
论文摘要
稳态是指系统的输出$ x_o $在输入$ \ Mathcal {i} $的变化上大约是恒定的。稳态经常出现在生化网络和其他相互作用元素网络中，其中数学模型基于与网络相关的微分方程。可以将这些网络抽象为digraphs $ \ mathcal {g} $，带有杰出的输入节点$ b，不同的杰出输出节点$ o $和许多监管节点$ρ_1，\ ldots，ρ_n$。在这些模型中，输入输出映射$ x_o（\ mathcal {i}）$由稳定的平衡$ x_0 $以$ \ mathcal {i} _0 $定义。稳定性意味着每个$ \ mathcal {i} $接近$ \ mathcal {i} _0 $ and nsimal sometasisis出现在$ \ mathcal {i} _0 $时，都有一个稳定的平衡$ x（\ mathcal {i}）$ $（dx_o/d \ mathcal {i}）（\ Mathcal {i} _0）= 0 $。我们表明，有一个$（n+1）\ times（n+1）$ stolotasis矩阵$ h（\ mathcal {i}）$，其中$ dx_o/d \ mathcal {i} = 0 $ if and of $ \ det（h）= 0 $。我们注意到，$ h $中的条目是线性化的耦合，$ \ det（h）$是这些条目中$ n+1 $的同质多项式。我们使用组合矩阵理论来考虑多项式$ \ det（H）$，从而确定与每个Digraph $ \ Mathcal {G} $相关的不同类型的可能体内平衡的菜单。具体来说，我们证明每个因素对应于$ \ Mathcal {G} $的子网。这些因素分为两个组合定义的类别：结构和附属物。结构因子对应于前馈基序和附属因素对应于反馈基序。最后，我们发现了一种用于确定与$ \ det（h）$的每个因子相对应的稳态子网基序的算法，而无需对模型方程进行数值模拟。该算法使我们能够对$ \ det（h）$的低度因子进行分类。
Homeostasis refers to a phenomenon whereby the output $x_o$ of a system is approximately constant on variation of an input $\mathcal{I}$. Homeostasis occurs frequently in biochemical networks and in other networks of interacting elements where mathematical models are based on differential equations associated to the network. These networks can be abstracted as digraphs $\mathcal{G}$ with a distinguished input node $ι$, a different distinguished output node $o$, and a number of regulatory nodes $ρ_1,\ldots,ρ_n$. In these models the input-output map $x_o(\mathcal{I})$ is defined by a stable equilibrium $X_0$ at $\mathcal{I}_0$. Stability implies that there is a stable equilibrium $X(\mathcal{I})$ for each $\mathcal{I}$ near $\mathcal{I}_0$ and infinitesimal homeostasis occurs at $\mathcal{I}_0$ when $(dx_o/d\mathcal{I})(\mathcal{I}_0) = 0$. We show that there is an $(n+1)\times(n+1)$ homeostasis matrix $H(\mathcal{I})$ for which $dx_o/d\mathcal{I} = 0$ if and only if $\det(H) = 0$. We note that the entries in $H$ are linearized couplings and $\det(H)$ is a homogeneous polynomial of degree $n+1$ in these entries. We use combinatorial matrix theory to factor the polynomial $\det(H)$ and thereby determine a menu of different types of possible homeostasis associated with each digraph $\mathcal{G}$. Specifically, we prove that each factor corresponds to a subnetwork of $\mathcal{G}$. The factors divide into two combinatorially defined classes: structural and appendage. Structural factors correspond to feedforward motifs and appendage factors correspond to feedback motifs. Finally, we discover an algorithm for determining the homeostasis subnetwork motif corresponding to each factor of $\det(H)$ without performing numerical simulations on model equations. The algorithm allows us to classify low degree factors of $\det(H)$.
