Computational analysis of synergism in small networks with different logic

Abstract

Cell fate decision processes are regulated by networks which contain different molecules and interactions. Different network topologies may exhibit synergistic or antagonistic effects on cellular functions. Here, we analyze six most common small networks with regulatory logic AND or OR, trying to clarify the relationship between network topologies and synergism (or antagonism) related to cell fate decisions. We systematically examine the contribution of both network topologies and regulatory logic to the cell fate synergism by bifurcation and combinatorial perturbation analysis. Initially, under a single set of parameters, the synergism of three types of networks with AND and OR logic is compared. Furthermore, to consider whether these results depend on the choices of parameter values, statistics on the synergism of five hundred parameter sets is performed. It is shown that the results are not sensitive to parameter variations, indicating that the synergy or antagonism mainly depends on the network topologies rather than the choices of parameter values. The results indicate that the topology with "Dual Inhibition" shows good synergism, while the topology with "Dual Promotion" or "Hybrid" shows antagonism. The results presented here may help us to design synergistic networks based on network structure and regulation combinations, which has promising implications for cell fate decisions and drug combinations.

Keywords: Cell fate decisions; Logic gates; Network topologies; Synergism.

Fig. 1

Different network topologies with three nodes. a “Dual Inhibition”. b “Dual Promotion”. c “Hybrid”

Fig. 2

Synergism analysis of the “Dual Inhibition” network in Fig. 1a(1). a and b Single bifurcation diagrams with AND and OR logic when $K_b$ or $K_c$ is individually perturbed, respectively. $K_{\mathit{bb}}$ and $K_{\mathit{cb}}$ refer to the basal values of $K_b$ and $K_c$ at which they are perturbed. The red and blue solid curves represent steady states of B with AND logic and OR logic, respectively, while black dotted lines indicate unstable states (This is true for all graphics below). c Two-parameter bifurcation diagrams with $K_b$ and $K_c$ as control parameters. The red and blue dashed curves represent saddle-node bifurcation points on the right side of the AND and OR logic bifurcation diagrams, i.e., $\mathrm{S}{N}_2$ and $\mathrm{S}{N}_4$. The red and blue solid curves represent the left saddle-node points of the AND and OR logic bifurcation diagrams, i.e., $\mathrm{S}{N}_1$ and $\mathrm{S}{N}_3$. d The normalized right bifurcation curves of AND and OR logic all locate below the isobole, which means combinatorial perturbations to $K_b$ and $K_c$ are synergistic due to ${\mathrm{K}}_{\mathit{bC}}/\mathrm{\Delta }{K}_b^{\ast }+{\mathrm{K}}_{\mathit{cC}}/\mathrm{\Delta }{K}_c^{\ast }=CI<1$. $K_{\mathit{bC}}/\mathrm{\Delta }{K}_b^{\ast }$ and $K_{\mathit{cC}}/\mathrm{\Delta }{K}_c^{\ast }$ represent the ratio required for state transition caused by combinatorial perturbations

Fig. 3

Synergism analysis of the “Dual Inhibition” network in Fig. 1a(2). a and b Single parameter bifurcation diagrams with AND and OR logic when $K_b$ or $K_c$ is individually perturbed. c Two-parameter bifurcation diagrams with AND and OR logic when $K_b$ and $K_c$ are combinatorially perturbed. d The normalized right bifurcation curves of AND and OR logic locate below the isobole line, i.e., $K_{\mathit{bC}}/\mathrm{\Delta }{K}_b^{\ast }+K_{\mathit{cC}}/\mathrm{\Delta }{K}_c^{\ast }<1$

Fig. 4

Synergism analysis of the “Dual Promotion” network in Fig. 1b(1). a and b Single parameter bifurcation diagrams of AND and OR logic with $K_b$ or $K_c$ as control parameter. c Two-parameter bifurcation diagrams of AND and OR logic when $K_b$ and $K_c$ are perturbed combinatorially. d The normalized left bifurcation curves locate above the isobole line, which indicates that the combinatorial perturbations to $K_b$ and $K_c$ are antagonistic, i.e., $K_{\mathit{bC}}/\mathrm{\Delta }{K}_b^{\ast }+K_{\mathit{cC}}/\mathrm{\Delta }{K}_c^{\ast }>1$

Fig. 5

Synergism analysis of the “Dual Promotion” network in Fig. 1b(2). a and b Single parameter bifurcation diagrams of AND and OR logic by perturbing $K_b$ or $K_c$ separately. c Two-parameter bifurcation diagrams of AND and OR logic by perturbing $K_b$ and $K_c$ combinatorially. d The normalized left bifurcation curves of AND and OR logic are all located above the isobole, which means that the combinatorial perturbation of $K_b$ and $K_c$ is antagonistic, i.e., ${\mathrm{K}}_{\mathit{bC}}/\mathrm{\Delta }{K}_b^{\ast }+{\mathrm{K}}_{\mathit{cC}}/\mathrm{\Delta }{K}_c^{\ast }=CI>1$

Fig. 6

Synergism analysis of the “Hybrid” network in Fig. 1c(1). a and b Bifurcation diagrams of AND and OR logic by perturbing $K_b$ or $K_c$ separately. c Two-parameter bifurcation diagrams of AND and OR logic by perturbing $K_b$ and $K_c$ combinatorially. d The normalized bifurcation curves of AND and OR logic all locate above the isobole, which means that combinatorial perturbations to $K_b$ and $K_c$ are antagonistic, i.e., $K_{\mathit{bC}}/\mathrm{\Delta }{K}_b^{\ast }+K_{\mathit{cC}}/\mathrm{\Delta }{K}_c^{\ast }>1$

Fig. 7

Synergism analysis of the “Hybrid’ network in Fig. 1c(2). a and b Bifurcation diagrams of AND and OR logic by perturbing $K_b$ or $K_c$ separately. c Two-parameter bifurcation diagrams of AND and OR logic by perturbing $K_b$ and $K_c$ combinatorially. d The normalized bifurcation curves of AND and OR logic all locate above the isobole, which means that combinatorial perturbations to $K_b$ and $K_c$ are antagonistic, i.e., $K_{\mathit{bC}}/\mathrm{\Delta }{K}_b^{\ast }+K_{\mathit{cC}}/\mathrm{\Delta }{K}_c^{\ast }>1$

Fig. 8

Distribution of CI for AND and OR logic in “Dual Inhibition” networks. Most CI values fall in the synergistic range, i.e., CI$<1$, indicating good synergism in the “Dual Inhibition” networks with two types of logic although the AND logic is much better

Fig. 9

Distribution of CI for AND and OR logic in “Dual Promotion” networks. The CI values under two types of logic are highly consistent in inducing antagonism

Fig. 10

Distribution of CI for AND and OR logic in “Hybrid” networks. Both types of logic in “Hybrid” networks are antagonistic

Fig. 11

The bistable switching bifurcation diagrams with $a_1$ and $a_2$ as govern parameters are given. $(\mathbf{a})$ The system produces saddle-node bifurcation by separately perturbing the parameter $a_1$. This $\mathrm{\Delta }{a}_1^{\ast }$ is the critical disturbance required from the basic value $a_{1b}$ to the saddle-node bifurcation value, which realizes the switch from low steady state AL to high steady state AH. The solid and dashed lines indicate steady state and unstable state respectively. $(\mathbf{b})$ The system generates a saddle-node bifurcation graph by individually perturbing the parameter $a_2$, which is similar to the state transition produced by the perturbation $a_1$. $(\mathbf{c})$ The parameter space ($a_1$, $a_2$) generated by the simultaneous perturbation of the two parameters is shown. These $\mathbf{I}$ and $\mathbf{III}$ represent the monostable region and $\mathbf{II}$ represents the bistable region. $(\mathbf{d})$ A sample isobologram of $a_1$ and $a_2$ combination calculation. The dashed line overlapping with the isobole indicates that the combination and the single action have the same effect. The blue curve is concave (located in the magenta part), which indicates the combinatorial perturbation of these two links is synergistic, i.e., $a_{1C}/\mathrm{\Delta }{a}_1^{\ast }+a_{2C}/\mathrm{\Delta }{a}_2^{\ast }<1$. If the blue curve is convex (located in the light purple part), which indicates that the combinatorial regulation of these two intensities is antagonistic, i.e., $a_{1C}/\mathrm{\Delta }{a}_1^{\ast }+a_{2C}/\mathrm{\Delta }{a}_2^{\ast }>1$