Homeostasis in Networks with Multiple Input Nodes and Robustness in Bacterial Chemotaxis

Abstract

A biological system achieve homeostasis when there is a regulated quantity that is maintained within a narrow range of values. Here we consider homeostasis as a phenomenon of network dynamics. In this context, we improve a general theory for the analysis of homeostasis in network dynamical systems with distinguished input and output nodes, called `input-output networks'. The theory allows one to define `homeostasis types' of a given network in a `model independent' fashion, in the sense that the classification depends on the network topology rather than on the specific model equations. Each `homeostasis type' represents a possible mechanism for generating homeostasis and is associated with a suitable `subnetwork motif' of the original network. Our contribution is an extension of the theory to the case of networks with multiple input nodes. To showcase our theory, we apply it to bacterial chemotaxis, a paradigm for homeostasis in biochemical systems. By considering a representative model of Escherichia coli chemotaxis, we verify that the corresponding abstract network has multiple input nodes. Thus showing that our extension of the theory allows for the inclusion of an important class of models that were previously out of reach. Moreover, from our abstract point of view, the occurrence of homeostasis in the studied model is caused by a new mechanism, called input counterweight homeostasis. This new homeostasis mechanism was discovered in the course of our investigation and is generated by a balancing between the several input nodes of the network -- therefore, it requires the existence of at least two input nodes to occur. Finally, the framework developed here allows one to formalize a notion of `robustness' of homeostasis based on the concept of `genericity' from the theory dynamical systems. We discuss how this kind of robustness of homeostasis appears in the chemotaxis model.

Figures (9)

• Figure 1: Time series of the model \ref{['original_e_coli']}, showing perfect homeostasis of the three variables $a_p$ (green), $b_p$ (cyan), $y_p$ (blue) at the non-dimensional equilibrium given by $a_p^* = 5.5 \times 10^{-3}$, $b_p^* = 0.48$, $y_p^* = 0.41$. Input parameter $L$ is given by a step function (red curve). The parameters were set to non-dimensional values of edgington18. Time series were computed using the software XPPAutbard02.
• Figure 2: (a) A core network $\mathcal{G}$ with input nodes $\iota_{1}$ and $\iota_{2}$, and output node $o$ (see definition \ref{['defining_core_networks']}). (b) Simple paths between the input nodes and the output node. (c) Nodes $\iota_{1}$ and $\sigma_{1}$ (both in blue) are $\iota_{1}$-simple, but they are not downstream $\iota_{2}$; node $\iota_{2}$ is $\iota_{2}$-simple, but it is not downstream $\iota_{2}$. Nodes highlighted in purple are either absolutely simple ($\sigma_{2}$) or absolutely super-simple ($\sigma_{3}$ and $o$). Both $\alpha$ and $\beta$ (in green) are absolutely appendage nodes (definitions \ref{['combinatorial_definitions_in_network_multiple_input_nodes']} and \ref{['useful_definitions_structural_multiple_input_nodes']}). The algorithm of subsection \ref{['algorithm_core_network_multiple_input_nodes']}, implies that $\mathcal{G}$ supports three classes of homeostasis: appendage (d); structural (e); and input counterweight (f).
• Figure 3: A core network with input nodes $\iota_{1}$ and $\iota_{2}$ and output node $o$. The only absolutely super-simple node is $o$. Node $\sigma$ is an absolutely simple node, but it is not between two absolutely super-simple nodes, as it would be expected for core networks with only one input node (see Lemma \ref{['relation_i_m_simple_node_and_i_m_super_simple_node']}).
• Figure 4: The possible connections in $\mathcal{G}$. The corresponding core network $\mathcal{G}_{c}$ is composed by the input nodes $\iota_{1}, \ldots, \iota_{n}$, the nodes $\sigma$ that are upstream from the output node and downstream of at least one input node, and the output node $o$.
• Figure 5: (a) An abstract network $\mathcal{G}$ with output node $o$ and input nodes $\iota_{1}$ and $\iota_{2}$. (b) Nodes and arrows that belong to the core subnetwork $\mathcal{G}_{1}$ between $\iota_{1}$ and $o$ are highlighted in blue. (c) Nodes and arrows that belong to the core subnetwork $\mathcal{G}_{2}$ between $\iota_{2}$ and $o$ are highlighted in red. (d) The core network $\mathcal{G}_{c}$ is obtained by the union between $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$. Nodes and arrows of $\mathcal{G}_{c}$ that belong to both $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ are highlighted in purple. Nodes and arrows which belong to $\mathcal{G}_{1}$ but not to $\mathcal{G}_{2}$ are highlighted in blue and nodes and arrows which belong to $\mathcal{G}_{2}$ but not to $\mathcal{G}_{1}$ are highlighted in red.

Theorems & Definitions (89)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma \oldthetheorem
• proof
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Remark 2.8