Defining network topologies that can achieve biochemical adaptation

Abstract

Many signaling systems show adaptation-the ability to reset themselves after responding to a stimulus. We computationally searched all possible three-node enzyme network topologies to identify those that could perform adaptation. Only two major core topologies emerge as robust solutions: a negative feedback loop with a buffering node and an incoherent feedforward loop with a proportioner node. Minimal circuits containing these topologies are, within proper regions of parameter space, sufficient to achieve adaptation. More complex circuits that robustly perform adaptation all contain at least one of these topologies at their core. This analysis yields a design table highlighting a finite set of adaptive circuits. Despite the diversity of possible biochemical networks, it may be common to find that only a finite set of core topologies can execute a particular function. These design rules provide a framework for functionally classifying complex natural networks and a manual for engineering networks. For a video summary of this article, see the PaperFlick file with the Supplemental Data available online.

Figure 1. Searching Topology Space for Adaptation Circuits

(A) Input-output curve defining adaptation. (B) Possible directed links among three nodes. (C) Illustrative examples of three-node circuit topologies. (D) Illustration of the analysis procedure for a given topology.

Figure 2. Minimal Networks (≤3 Links) Capable of Adaptation

(A) Adaptive networks composed of negative feedback loops. Three examples of adaptation networks are shown in the upper panel. Each is one member (shaded) of a group of similar adaptation networks, whose signs of regulations are listed underneath. For comparison, three examples of nonadaptive networks are shown in the low panel, with their “defects” for adaptation function listed underneath. (B) Adaptive networks composed of incoherent feedforward loops. The only two minimal adaptation networks in this case are shown in the upper panel. Examples of nonadaptive networks are shown in the lower panel.

Figure 3. Phase Diagram and Nullcline Analysis of Representative Networks from the Two Classes of Minimal Adaptive Topologies

The two networks are shown on the top with the key regulations colored to indicate the parameter constraints for achieving perfect adaptation. (A) Phase planes of the variables B and C for a NFBLB topology. The B nullclines are drawn in black lines and C nullclines in red (solid red for the initial input I₁ and dashed red for the changed input I₂). The steady states with input I₁ and I₂ are the intersections of the nullclines and are highlighted by black and gray dots, respectively. When the input is changed from I₁ to I₂, the trajectory (blue lines) of the system variables follows the vector field (dB/dt, dC/dt) (with input I₂), which is denoted by the green arrows. The trajectory’s projection on the C axis is the system’s output and is shown separately right next to the phase plane. (Refer to Figure 1A for the functional meaning of O₁, O₂, andO_peak.) Two sets of key parameters (K_M’s on B) are used to illustrate their effect on adaptation precision: $K_{F_{0}B}^{\prime }=0.1$ and K_CB = 0.1 for the top panel and $K_{F_{0}B}^{\prime }=0.01$ and K_CB = 0.01 for the middle and lower panels. Two sets of rate constants are used to illustrate their effect on sensitivity: k_AC = 10 and k′_BC = 10 for the top and the middle panels and k_AC = 0.1 and k′_BC = 0.1 for the lower panel. (B) Phase planes for an IFFLP topology. $K_{F_{0}B}^{\prime }=1$ and K_AB = 0.1 for the top panel. $K_{F_{0}B}^{\prime }=100$ and K_AB = 0.001 for the middle and the lower panels. k_AB = 0.5 and $k_{F_{0}B}^{\prime }=10$ for the top and the middle panels. k_AB = 100 and $k_{F_{0}B}^{\prime }=2000$ for the lower panel.

Figure 4. Searching the Full Circuit-Space for All Robust Adaptation Networks

(A) The probability plot for all 16,038 networks with all the parameters sampled. Three hundred and ninety-five networks are overrepresented in the functional region shown by the orange rectangle. (B) Venn diagram of networks with three characters: adaptive, containing negative FBL, and containing incoherent FFL. (C) Clustering of the adaptation networks that belong to the NFBLB class. The network motifs associated with each of the subclusters are shown on the right. (D) Clustering of adaptation networks that belong to the IFFLP class.

Figure 5. General Analysis for Adaptive Circuits

(A) Relevant equations. The steady-state output change ΔC* with respect to the input change ΔI can be derived from the linearized steady-state equations. A zero adaptation error around a stable steady state requires a zero minor |N| and a nonzero determinant |J| < 0. There are two terms I and II in |N|, and |N| = 0 implies either both terms are zero or they are equal but nonzero. We are only interested in robust adaptation, i.e., the cases where the condition leading to |N| = 0 holds within a range of parameters and input values. (B) NFBLB class of adaptive circuits (I = II = 0). In this category ∂f_C/∂A ≠ 0, which means that there is always a link from node A to node C. (Otherwise, there would be no direct or indirect path from A to C.) Then II = 0 implies that ∂f_B/B is always zero. I = 0 implies that at least one of ∂f_B/∂A and ∂ f_C/∂B is zero. This condition implies that there is no feedforward loop in this category. In our enzymatic model ∂f_B/∂B = 0 can be robustly achieved either by saturating the enzymes on the node B so that f_B does not depend on B explicitly or by adding a positive self-loop on node B so that the dependence of f_B on B can be factored out. An example of the latter is when node B is regulated by itself positively and by C negatively, so that ${\mathrm{f}}_B=k_{\mathit{BB}}B\left(1-B\right)/\left(1-B+K_{\mathit{BB}}\right)-k_{\mathit{CB}}^{\prime }\mathit{CB}/\left(B+k_{\mathit{CB}}^{\prime }\right)\approx k_{\mathit{BB}}B-k_{\mathit{CB}}^{\prime }\mathit{CB}/k_{\mathit{CB}}^{\prime }=B\left(k_{\mathit{BB}}-k_{\mathit{CB}}^{\prime }/k_{\mathit{CB}}^{\prime }C\right)$, in the limits (1-B) ≫ K_BB and B ≪ K′ _CB. The terms in the determinant |J| correspond to different feedback loops as colored in the figure. Thus, there should be at least one, but can be two, negative feedback loops in this category. (C) IFFLP class (I = II ≠ 0). In this category, none of the factors in |N| are zero. This implies the presence of the links colored in the figure and hence a FFL. The condition for |N| = 0 can be robustly satisfied if the FFL exerts two opposing but proportional regulations on C. The proportionality relationship can be established by f_B taking the form shown in the figure.

Figure 6. Design Table of Adaptation Networks

Two examples are shown on the left for the NFBLB class of adaptation networks, which require a core NFBLB motif with the node B functioning as a buffer. One example is shown on the right for the IFFLP class, which require a core IFFLP motif with the node B functioning as a proportioner. The table is constructed by adding more and more beneficial motifs to the minimal adaptation networks. The Q value (Robustness) of each network is shown underneath, along with the mathematical relation the node B establishes.

Figure 7. The Network of Perfect Adaptation in E. coli Chemotaxis Belongs to the NFBLB Class of Adaptive Circuits

Left: the original network in E. coli. Middle: the redrawn network to highlight the role and the control of the key node “Methylation Level.” Right: one of the minimal adaptation networks in our study.