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.

Appendices

Appendix A

1.1 Formulation of logic gates

To identify the dependence of synergy on different logic, we enumerate six small networks, as shown in Fig. 1. Based on the different logic modeling methods (Section 4.2), ordinary differential equations of AND logic and OR logic for these six networks are given.

OR logic

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{A}=\frac{V_x}{(1+(B/K_a)^H)}-aA+D, &{} \\ \dot{B}=\frac{V_y}{(1+(A/K_b)^H+(C/K_c)^H)}-bB+E, &{} \\ \dot{C}=\frac{V_z}{(1+(B/K_d)^H)}-cC+F. \end{array} \right. \end{aligned}$$

(15)

AND logic

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{A}=\frac{V_x}{(1+(B/K_a)^H)}-aA+D, &{} \\ \dot{B}=\frac{V_y}{(1+(A/K_b)^H*(C/K_c)^H)}-bB+E, &{} \\ \dot{C}=\frac{V_z}{(1+(B/K_d)^H)}-cC+F. \end{array} \right. \end{aligned}$$

(16)

OR logic

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{A}=\frac{V_x((B/K_a)^H)}{(1+(B/K_a)^H)}-aA+D, &{} \\ \dot{B}=\frac{V_y}{(1+(A/K_b)^H+(C/K_c)^H)}-bB+E, &{} \\ \dot{C}=\frac{V_z}{(1+(B/K_d)^H)}-cC+F. \end{array} \right. \end{aligned}$$

(17)

AND logic

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{A}=\frac{V_x((B/K_a)^H)}{(1+(B/K_a)^H)}-aA+D, &{} \\ \dot{B}=\frac{V_y}{(1+(A/K_b)^H*(C/K_c)^H)}-bB+E, &{} \\ \dot{C}=\frac{V_z}{(1+(B/K_d)^H)}-cC+F. \end{array} \right. \end{aligned}$$

(18)

OR logic

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{A}=\frac{V_x((B/K_a)^H))}{(1+(B/K_a)^H)}-aA+D, &{} \\ \dot{B}=\frac{V_y((A/K_b)^H+(C/K_c)^H)}{(1+(A/K_b)^H+(C/K_c)^H)}-bB+E, &{} \\ \dot{C}=\frac{V_z((B/K_d)^H)}{(1+(B/K_d)^H)}-cC+F. \end{array} \right. \end{aligned}$$

(19)

AND logic

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{A}=\frac{V_x((B/K_a)^H)}{(1+(B/K_a)^H)}-aA+D, &{} \\ \dot{B}=\frac{V_y((A/K_b)^H*(C/K_c)^H)}{(1+(A/K_b)^H*(C/K_c)^H)}-bB+E, &{} \\ \dot{C}=\frac{V_z((B/K_d)^H)}{(1+(B/K_d)^H)}-cC+F. \end{array} \right. \end{aligned}$$

(20)

OR logic

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{A}=\frac{V_x((B/K_a)^H)}{(1+(B/K_a)^H)}-aA+D, &{} \\ \dot{B}=\frac{V_y((A/K_b)^H+(C/K_c)^H)}{(1+(A/K_b)^H+(C/K_c)^H)}-bB+E, &{} \\ \dot{C}=\frac{V_z}{(1+(B/K_d)^H)}-cC+F. \end{array} \right. \end{aligned}$$

(21)

AND logic

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{A}=\frac{V_x((B/K_a)^H)}{(1+(B/K_a)^H)}-aA+D, &{} \\ \dot{B}=\frac{V_y((A/K_b)^H*(C/K_c)^H)}{(1+(A/K_b)^H*(C/K_c)^H)}-bB+E, &{} \\ \dot{C}=\frac{V_z}{(1+(B/K_d)^H)}-cC+F. \end{array} \right. \end{aligned}$$

(22)

OR logic

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{A}=\frac{V_x}{(1+(B/K_a)^H)}-aA+D, &{} \\ \dot{B}=\frac{V_y((C/K_c)^H)}{(1+(A/K_b)^H+(C/K_c)^H)}-bB+E, &{} \\ \dot{C}=\frac{V_z((B/K_d)^H)}{(1+(B/K_d)^H)}-cC+F. \end{array} \right. \end{aligned}$$

(23)

AND logic

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{A}=\frac{V_x}{(1+(B/K_a)^H)}-aA+D, &{} \\ \dot{B}=\frac{V_y((C/K_c)^H)}{(1+(A/K_b)^H*(C/K_c)^H)}-bB+E, &{} \\ \dot{C}=\frac{V_z((B/K_d)^H)}{(1+(B/K_d)^H)}-cC+F. \end{array} \right. \end{aligned}$$

(24)

OR logic

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{A}=\frac{V_x((B/K_a)^H))}{(1+(B/K_a)^H)}-aA+D, &{} \\ \dot{B}=\frac{V_y((C/K_c)^H)}{(1+(A/K_b)^H+(C/K_c)^H)}-bB+E, &{} \\ \dot{C}=\frac{V_z((B/K_d)^H)}{(1+(B/K_d)^H)}-cC+F. \end{array} \right. \end{aligned}$$

(25)

AND logic

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{A}=\frac{V_x((B/K_a)^H)}{(1+(B/K_a)^H)}-aA+D, &{} \\ \dot{B}=\frac{V_y((C/K_c)^H)}{(1+(A/K_b)^H*(C/K_c)^H)}-bB+E, &{} \\ \dot{C}=\frac{V_z((B/K_d)^H)}{(1+(B/K_d)^H)}-cC+F. \end{array} \right. \end{aligned}$$

(26)

Here, variables A, B,and C represent the concentration of the three proteins at time t. The parameters used for the following calculations are listed in Table 5. All the bifurcation diagrams in the above text are drawn by software Matcont (a subpackage available under Matlab) [52], and all random and statistical processes are completed by Matlab.

Table 5 Parameters are used for simulation and calculation

Appendix B

1.1 Steady-state comparison of OR logic and AND logic

In this section, the steady state of AND and OR logic in Fig. 1a(1) are compared by theoretical calculation. Models (15) and (16) are used to calculate, $K_b$ is regarded as the bifurcation parameter. Here, $(A^*_{OR}, B^*_{OR}, C^*_{OR})$ and $(A^*_{AND}, B^*_{AND}, C^*_{AND})$ denotes the low-state equilibrium point of the OR logic and AND logic. $(\tilde{A}_{OR}, \tilde{B}_{OR}, \tilde{C}_{OR})$ and $(\tilde{A}_{AND}, \tilde{B}_{AND}, \tilde{C}_{AND})$ denotes the high-state equilibrium point of the OR logic and AND logic.

First, the low steady state of OR logic is given. From Eq. (15), we know that when $K_b$ $\rightarrow$0, the variable B$\rightarrow$0, i.e., B is in the low state. The A and C can be approximated

$$\begin{aligned} \begin{array}{ll} A\rightarrow \frac{V_x}{a},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C\rightarrow \frac{V_z}{c}, \end{array} \end{aligned}$$

(27)

Substituting Eq. (27) into the formula B in model (15), we get

$$\begin{aligned} \begin{array}{ll} B^*_{OR}=\frac{V_y}{b\big [1+(\frac{V_x}{a K_b})^2+(\frac{V_z}{c K_c})^2\big ]} \end{array} \end{aligned}$$

(28)

Substituting Eq. (28) into model (15), equilibrium points $A^*_{OR}$ and $C^*_{OR}$ can be obtained

$$\begin{aligned} \begin{array}{ll} A^*_{OR}=\frac{V_x}{a\left\{ 1+\left( \frac{V_y}{K_a b\big [1+(\frac{V_x}{a K_b})^2+(\frac{V_z}{c K_c})^2\big ]}\right) ^2\right\} }, \end{array} \end{aligned}$$

(29)

$$\begin{aligned} \begin{array}{ll} C^*_{OR}=\frac{V_z}{c\left\{ 1+\left( \frac{V_y}{K_d b\big [1+(\frac{V_x}{a K_b})^2+(\frac{V_z}{c K_c})^2\big ]}\right) ^2\right\} }. \end{array} \end{aligned}$$

(30)

For AND logic, substituting Eq. (27) into the formula B in model (16), we get

$$\begin{aligned} \begin{array}{ll} B^*_{AND}=\frac{V_y}{b\big [1+(\frac{V_x}{a K_b})^2*(\frac{V_z}{c K_c})^2\big ]} \end{array} \end{aligned}$$

(31)

Substituting Eq. (31) into model (16), equilibrium points $A^*_{AND}$ and $C^*_{AND}$ can be obtained

$$\begin{aligned} \begin{array}{ll} A^*_{AND}=\frac{V_x}{a\left\{ 1+\left( \frac{V_y}{K_a b\big [1+(\frac{V_x}{a K_b})^2*(\frac{V_z}{c K_c})^2\big ]}\right) ^2\right\} }, \end{array} \end{aligned}$$

(32)

$$\begin{aligned} \begin{array}{ll} C^*_{AND}=\frac{V_z}{c\left\{ 1+\left( \frac{V_y}{K_d b\big [1+(\frac{V_x}{a K_b})^2*(\frac{V_z}{c K_c})^2\big ]}\right) ^2\right\} }. \end{array} \end{aligned}$$

(33)

When $K_b$ is in the low state, the values of formulas (28) and (31) can be used to compare the B low steady state of OR logic and AND logic. The parameter values are replaced in Table 5, $B_{OR}>B_{AND}$ can be obtained, which shows good consistency with Fig. 2.

Next, the high steady state of OR logic and AND logic in Fig. 1a(1) are compared. For OR logic, when $K_b\rightarrow \infty$, $B\rightarrow \frac{V_y}{b\left( 1+\left( \frac{C}{K_c}\right) ^2\right) }$. B is substituted into Eq. (15), we can obtain

$$\begin{aligned} \begin{array}{ll} \frac{V_x}{(1+(B/K_a)^2)}-aA=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \frac{V_z}{(1+(B/K_d)^2)}-cC=0, \end{array} \end{aligned}$$

(34)

The root of A and C can be determined by Eq. (34). Substituting the roots into model (15), we can get

$$\begin{aligned} \begin{array}{ll} \tilde{B}_{OR}=\frac{V_y}{b(1+(A/K_b)^2+(C/K_c)^2)}, \end{array} \end{aligned}$$

(35)

Substituting Eq. (35) into model (15), $\tilde{A}_{OR}$ and $\tilde{C}_{OR}$ can be obtained

$$\begin{aligned} \begin{array}{ll} \tilde{A}_{OR}=\frac{V_x}{a\left[ 1+\left( \frac{\tilde{B}_{OR}}{K_a}\right) ^2\right] }, \end{array} \end{aligned}$$

(36)

$$\begin{aligned} \begin{array}{ll} \tilde{C}_{OR}=\frac{V_z}{c\left[ 1+\left( \frac{\tilde{B}_{OR}}{K_d}\right) ^2\right] }, \end{array} \end{aligned}$$

(37)

For AND logic, when $K_b\rightarrow \infty$, $B\rightarrow \frac{V_y}{b}$. A and C is obtained

$$\begin{aligned} \begin{array}{ll} A\rightarrow \frac{V_x(K_a b)^2}{a(V_y^2+(K_a b)^2)},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C\rightarrow \frac{V_z(K_d b)^2}{c(V_y^2+(K_d b)^2)}, \end{array} \end{aligned}$$

(38)

Substituting Eq. (38) into the formula B of model (16), we can get

$$\begin{aligned} \begin{array}{ll} \tilde{B}_{AND}=\frac{V_y}{b\left\{ 1+\left( \left( \frac{V_x(b Ka)^2}{a K_b((b K_a)^2+V_y^2)}\right) ^2*\left( \frac{V_z(b K_d)^2}{c K_c((b K_d)^2+V_y^2)}\right) \right) ^2\right\} }, \end{array} \end{aligned}$$

(39)

Substituting Eq. (39) into model (16), equilibrium points $\tilde{A}_{AND}$ and $\tilde{C}_{AND}$ are obtained

$$\begin{aligned} \begin{array}{ll} A=\frac{V_x}{a\left\{ 1+\left[ \frac{V_y}{K_a b\left( 1+\left( \frac{V_x(b K_a)^2}{a K_b((b K_a)^2+V_y^2)}\right) ^2*\left( \frac{V_z(b K_d)^2}{c K_c((b K_d)^2+V_y^2)}\right) ^2\right) }\right] ^2\right\} }, \end{array} \end{aligned}$$

(40)

$$\begin{aligned} \begin{array}{ll} C=\frac{V_z}{c\left\{ 1+\left[ \frac{V_y}{K_d b\left( 1+\left( \frac{V_x(b Ka)^2}{a K_b((b K_a)^2+V_y^2)}\right) ^2*\left( \frac{V_z(b K_d)^2}{c K_c((b K_d)^2+V_y^2)}\right) ^2\right) }\right] ^2\right\} }, \end{array} \end{aligned}$$

(41)

When $K_b$ is in high steady state, Eqs. (35) and (39) can be used to compare the high steady state of OR logic and AND logic. $K_b$ takes a larger value, and other values are replaced by Table 5, the calculation result is $B_{AND}>B_{OR}$, which is consistent with Fig. 2.

1.2 Saddle-node bifurcation analysis of OR logic and AND logic

In this section, the bifurcation points of OR logic and AND logic are analyzed. Here, $K_b$ is regarded as a bifurcation parameter. First, the bifurcation point of OR logic (i.e., model (15)), is calculated. It assumed that $(A^*, B^*, C^*)$ is the equilibrium point of the bifurcation point of $K_b$.

Let $\bar{A}=A-A^*$, $\bar{B}=B-B^*$ and $\bar{C}=C-C^*$. $\bar{A}$, $\bar{B}$ and $\bar{C}$ are still represented by A, B and C respectively. Therefore, the linear form of the system (15) is expressed as follows

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{A}=-a A-Q B, &{} \\ \dot{B}=-R A-b B-SC, &{} \\ \dot{C}=-TB-c C, &{} \\ \end{array} \right. \end{aligned}$$

(42)

where

$$\begin{aligned}&Q=\frac{2(1/K_a)^2aA^*B^*}{1+(B^*/K_a)^2}, \\&R=\frac{2V_y(1/K_b)^2A^*}{(1+(A^*/K_b)^2+(C^*/K_c)^2)^{2}}, \\&S=\frac{2V_y(1/K_c)^2C^*}{(1+(A^*/K_b)^2+(C^*/K_c)^2)^{2}}, \\&T=\frac{2(1/K_d)^2cB^*C^*}{1+(B^*/K_d)^2}, \end{aligned}$$

where

$$\begin{aligned}&A^*=\frac{V_x}{a(1+(B^*/K_a)^2)}, \\&C^*=\frac{V_z}{c(1+(B^*/K_d)^2)}. \end{aligned}$$

Therefore, the Jacobian matrix of system (15) is

$$J= {\begin{vmatrix}-a &{} -Q &{} 0 \\ -R &{} -b &{} -S \\ 0 &{} -T &{} -c \end{vmatrix}}.$$

(43)

The characteristic equation of the linearized system (15) is

$${\begin{vmatrix} \lambda +a &{} Q &{} 0 \\ R &{} \lambda +b &{} S \\ 0 &{} T &{} \lambda +c \end{vmatrix} =0.}$$

(44)

Then, the exponential polynomial equation is obtained

$$\begin{aligned} \begin{array}{ll} \lambda ^{3}+(a+b+c)\lambda ^{2}+(ab+ac+bc-QR-ST)\lambda +(abc-cQR-aST)=0, \end{array} \end{aligned}$$

(45)

As we all know, when the system produces saddle-node bifurcation, at least one of all eigenvalues is zero. Therefore, substituting $\lambda =0$ into the polynomial Eq. (45), we can get

$$\begin{aligned} \begin{array}{ll} abc-cQR-aST=0, \end{array} \end{aligned}$$

(46)

The next goal is to solve the bifurcation point $K_b$ and equilibrium point $B^*$. From Eqs. (46) and (15), we get

$$\begin{aligned} \left\{ \begin{array}{ll} abc-cQR-aST=0, &{} \\ \frac{V_y}{1+\left( A^*/K_b\right) ^2+\left( C^*/K_c\right) ^2}-bB^*=0. \end{array} \right. \end{aligned}$$

(47)

Then, substituting all the parameters in Table 5 except the control parameter $K_b$ into Eq. (47). The bifurcation point $K_b$ and the equilibrium point $B^*$ can be obtained, i.e., ($K_b=1.30864$, $B^*=1.91313$) and ($K_b=1.56867$, $B^*=0.917858$), which are consistent with the blue curve in Fig. 2a and b.

Next, the bifurcation point of AND logic (i.e., model (16)), is analyzed. Here, it is supposed that $(\widetilde{A}, \widetilde{B}, \widetilde{C})$ is the equilibrium point of the bifurcation point $K_b$.

Let $\bar{A}=A-\widetilde{A}$, $\bar{B}=B-\widetilde{B}$ and $\bar{C}=C-\widetilde{C}$. $\bar{A}$, $\bar{B}$ and $\bar{C}$ are still represented by A, B, C. Therefore, the linear form of the system (16) is described as follows

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{A}=-a A-M B, &{} \\ \dot{B}=-H A-b B-WC, &{} \\ \dot{C}=-N B-c C, &{} \\ \end{array} \right. \end{aligned}$$

(48)

where

$$\begin{aligned}&M=\frac{2(1/K_a)^2 a\widetilde{B}\widetilde{A}}{1+(\widetilde{B}/K_a)^2}, \\&H=\frac{2V_y(1/K_b)^2(\widetilde{C}/K_c)^2\widetilde{A}}{(1+(\widetilde{A}/K_b)^2*(\widetilde{C}/K_c)^2)^{2}}, \\&W=\frac{2V_y(1/K_c)^2(\widetilde{A}/K_b)^2\widetilde{C}}{(1+(\widetilde{A}/K_b)^2*(\widetilde{C}/K_c)^2)^{2}}, \\&N=\frac{2(1/K_d)^2c\widetilde{B}\widetilde{C}}{1+(\widetilde{B}/K_d)^2}, \end{aligned}$$

where

$$\begin{aligned}&\widetilde{A}=\frac{V_x}{a(1+(\widetilde{B}/K_a)^2)}, \\&\widetilde{C}=\frac{V_z}{c(1+(\widetilde{B}/K_d)^2)}. \end{aligned}$$

Therefore, the Jacobian matrix of system (16) is

$${J= {}\begin{vmatrix} -a &{} -M &{} 0 \\ -H &{} -b &{} -W \\ 0 &{} -N &{} -c \end{vmatrix}.}$$

(49)

The characteristic equation of the linearized system (15) is

$${\begin{vmatrix} \lambda +a &{} M &{} 0 \\ H &{} \lambda +b &{} W \\ 0 &{} N &{} \lambda +c \end{vmatrix} =0.}$$

(50)

Then, the exponential polynomial equation is obtained

$$\begin{aligned} \begin{array}{ll} \lambda ^{3}+(a+b+c)\lambda ^{2}+(ab+ac+bc-MH-WN)\lambda +(abc-cMH-aWN)=0, \end{array} \end{aligned}$$

(51)

Substituting this zero eigenvalue into Eq. (51), the following equation is obtained

$$\begin{aligned} \begin{array}{ll} abc-cMH-aWN=0, \end{array} \end{aligned}$$

(52)

From Eqs. (16) and (52), we get

$$\begin{aligned} \left\{ \begin{array}{ll} abc-cMH-aWN=0, &{} \\ \frac{V_y}{1+\left( \widetilde{A}/K_b\right) ^2*\left( \widetilde{C}/K_c\right) ^2}-b\widetilde{B}=0. \end{array} \right. \end{aligned}$$

(53)

Substituting all the parameters in Table 5 except the control parameter $K_b$ into Eq. (53). The bifurcation point $K_b$ and the equilibrium point $\widetilde{B}$ of the AND logic are obtained, i.e., ($K_b$=0.366907, $\widetilde{B}$= 2.53285) and ($K_b$= 1.44897, $\widetilde{B}$= 0.5), which are consistent with red curve in Fig. 2a and b.