Robust network topologies for generating switch-like cellular responses

Abstract

Signaling networks that convert graded stimuli into binary, all-or-none cellular responses are critical in processes ranging from cell-cycle control to lineage commitment. To exhaustively enumerate topologies that exhibit this switch-like behavior, we simulated all possible two- and three-component networks on random parameter sets, and assessed the resulting response profiles for both steepness (ultrasensitivity) and extent of memory (bistability). Simulations were used to study purely enzymatic networks, purely transcriptional networks, and hybrid enzymatic/transcriptional networks, and the topologies in each class were rank ordered by parametric robustness (i.e., the percentage of applied parameter sets exhibiting ultrasensitivity or bistability). Results reveal that the distribution of network robustness is highly skewed, with the most robust topologies clustering into a small number of motifs. Hybrid networks are the most robust in generating ultrasensitivity (up to 28%) and bistability (up to 18%); strikingly, a purely transcriptional framework is the most fragile in generating either ultrasensitive (up to 3%) or bistable (up to 1%) responses. The disparity in robustness among the network classes is due in part to zero-order ultrasensitivity, an enzyme-specific phenomenon, which repeatedly emerges as a particularly robust mechanism for generating nonlinearity and can act as a building block for switch-like responses. We also highlight experimentally studied examples of topologies enabling switching behavior, in both native and synthetic systems, that rank highly in our simulations. This unbiased approach for identifying topologies capable of a given response may be useful in discovering new natural motifs and in designing robust synthetic gene networks.

Figure 1. Switch-like behavior.

A. A typical Michaelian system (n_H = 1) requires an 81-fold increase in stimulus to increase the response from 10% to 90% of the maximum (i.e., S_90%/S_10% = 81) while an ultrasensitive response is more abrupt. B. Once triggered into the high, or ‘on’, state (S>S_on), a bistable system stays in that state even as the stimulus concentration is decreased, only switching ‘off’ below a lower threshold stimulus concentration (S_off, which is <0 for irreversible systems).

Figure 2. Topology search scheme.

A. Each component is modeled as an enzyme or transcription factor. The input component A is modeled as a receptor to which the stimulus binds. B. Enzymatic components can catalyze the activation or inactivation of their targets, denoted as X. Transcriptional components can upregulate or inhibit the synthesis of the inactive forms of their targets. C. Sample network illustrating all possible interaction types. D. Four compositional classes were studied: EEE, in which A, B, C, are modeled as enzymes; TTT, in which each component is a transcription factor; and hybrid networks, in which only C is a transcription factor (EET) or both B and C are transcription factors (ETT). E. Overview of the topology search algorithm.

Figure 3. Robustness in switch-like behavior across compositional classes.

A. All possible network topologies were constructed and simulated; response profiles were used to compute robustness scores for ultrasensitivity and bistability for each network topology. This process was repeated for each compositional class. Histograms depict the distribution of robustness scores for ultrasensitivity and bistability greater than 1% across all compositional classes; white bars with oblique lines in the TTT plots depict the distribution of robustness scores when each transcriptional interaction is modeled as being cooperative (n_H = 2). Histograms represent ultrasensitivity robustness scores for EEE (226 networks), EET (699), ETT (1511), TTT (84), TTT n_H = 2 (1360) and bistability robustness scores for EEE (119 networks), EET (468), ETT (972), TTT (0), TTT n_H = 2 (43). Networks achieving the highest robustness scores belong to the hybrid classes: the most robust networks in the ETT class achieve the highest scores for both ultrasensitivity and bistabiltiy, and the most robust networks in EET achieve comparably high scores. B. Ultrasensitivity and bistability robustness scores for two example topologies under different compositional classes; the same network topology can yield dramatically different robustness scores under different compositional classes.

Figure 4. Ultrasensitivity via linear transcriptional feedback and degradation.

A simple linear transcriptional feedback system can give rise to ultrasensitivity even in the absence of an inactivating enzyme. Note that this figure pertains to simulations on a minimal model different from the setup used for the topology search simulations (see Methods). A. In this system, the transcription factor C is activated by an enzyme, A. C is subject to basal synthesis and first-order degradation, but not to inactivation. B. The model was simulated on 10⁶ random parameter sets, and a random subset of the results was plotted. Each dot represents a separate simulation on a random parameter set, and the color of the dot denotes the value of the dimensionless ratio in that parameter set (where b is the basal synthesis rate and v is the maximal feedback synthesis rate). If is sufficiently high, then the Hill coefficient reaches a maximum when the effective feedback synthesis rate constant (where K_F is the threshold concentration) is approximately equal to the degradation rate constant k_deg.

Figure 5. Minimal architecture for generating robust ultrasensitivity.

Starting with a simple network, incremental addition of specific interactions significantly improves robustness in generating ultrasensitivity. The map to the right lists the eight most robust network topologies generating ultrasensitivity in the EET class, after pruning; positive, negative, and no interactions are depicted with green, red, and black, respectively.

Figure 6. Coupling of ultrasensitive activation and positive synthesis feedback yields robust bistability.

The upper row depicts molecular mechanisms derived from simulation results and the lower row depicts concordant examples in oocyte maturation. In our simulations, ultrasensitivity can arise via zero-order effects, enzyme cascading, and linear synthesis feedback. These motifs can yield bistability when coupled with positive synthesis feedback, and multiple feedbacks contribute to the robustness of this bistability. The map to the right lists the eight most robust network topologies generating bistability in the ETT class.

Figure 7. Comparison with natural and synthetic systems.

A. Yan is a critical regulator of differentiation pathways in development, and generates ultrasensitivity via zero-order effects. B. The EpoR/GATA1 receptor/transcription factor pair can generate ultrasensitivity critical to the regulation of commitment to the erythrocytic lineage; this network is architecturally the same as the highest ranking network depicted in Fig. 5. C. The synthetic AtCRE1/SKN7 hybrid network depicted exhibits robust switch-like behavior in yeast. This network is architecturally the same as those in Figs. 5 and 7B.