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Functional characteristics of additional positive feedback in genetic circuits

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Abstract

Multiple-positive feedback circuits are ubiquitous regulatory motifs in complex bio-molecular networks. A popular topic is why multiple-positive feedback mechanisms have been evolved and selected by organisms. To this end, a two-component dual-positive feedback genetic circuit is investigated, which consists of an auto-activation loop and a double negative feedback circuit. The auto-activation loop acts as an additional positive feedback loop (APFL), and our aim is to explore the functional characteristics of the APFL. Investigations reveal that the APFL can regulate the size of bistable region and the robust attractiveness of stable steady states. It is also found that the APFL can regulate global relative input–output sensitivities of the system. Furthermore, the APFL can tune the response speed, noise resistance and stochastic switch behavior of the system, which makes it easy to realize functional tunability and robust decision-making. Therefore, rationalizing why multiple-positive feedback circuits so frequently appear in real-world biological systems. Potential applications of the associated investigations include the design of artificial genetic circuits, the modeling and model reduction for large-scale bio-molecular networks.

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Acknowledgments

The authors would like to thank Profs. Tianshou Zhou and Xiaoqun Wu for their valuable comments. This work is supported by the National Science and Technology Major Project of China under Grant 2014ZX10004-001-014, the 973 Project under Grant 2014CB845302, and the National Natural Science Foundation of China under Grants 61304151, 11105040, 61025017, 11472290, the Australia ARC Discovery Grants DP130104765, the Science Foundation of Henan University under Grants 2012YBZR007.

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Authors and Affiliations

  1. School of Mathematics and Information Sciences, Henan University, Kaifeng, 475004, People’s Republic of China

    Pei Wang & Yuhuan Zhang

  2. School of Electrical and Computer Engineering, RMIT University, Melbourne, 3001, VIC, Australia

    Pei Wang & Xinghuo Yu

  3. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Jinhu Lü

  4. Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia

    Jinhu Lü

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  1. Pei Wang
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Correspondence to Pei Wang.

Appendix: The derivation of mathematical model (1)

Appendix: The derivation of mathematical model (1)

From Table 1 and by the law of mass action, differential equations to describe the concentration evolutions of species in the fast equations can be obtained as follows:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\hbox {d}[X_2]}{\hbox {d}t}=k_1^{+}[X]^2-k_1^{-}[X_2]-k_2^{+}[X_2][S]+k_2^{-}[X^{*}],\\ \frac{\hbox {d}[X^{*}]}{\hbox {d}t}=k_2^{+}[X_2][S]-k_2^{-}[X^{*}]-k_3^{+}[X^{*}][D_y]+k_3^{-}[D_yX^{*}]\\ -k_6^{+}[X^{*}][D_x]+k_6^{-}[D_xX^{*}], \\ \frac{\hbox {d}[D_yX^{*}]}{\hbox {d}t}=k_3^{+}[X^{*}][D_y]-k_3^{-}[D_yX^{*}], \\ \frac{\hbox {d}[Y_2]}{\hbox {d}t}=k_4^{+}[Y]^2-k_4^{-}[Y_2]-k_5^{+}[Y_2][D_x]+k_5^{-}[D_xY_2],\\ \frac{\hbox {d}[D_xY_2]}{\hbox {d}t}=k_5^{+}[Y_2][D_x]-k_5^{-}[D_xY_2], \\ \frac{\hbox {d}[D_xX^{*}]}{\hbox {d}t}=k_6^{+}[X^{*}][D_x]-k_6^{-}[D_xX^{*}]. \\ \end{array}\right. \end{aligned}$$
(7)

Assume fast reactions can quickly reach their chemical equilibrium, one has: \([X_2]\!=\!K_1[X]^2,[X^{*}]\!=\!K_2[X_2][S], [D_yX^{*}] \!=\! K_3[X^{*}][D_y], [Y_2] \!\!=\! K_4[Y]^2, [D_xY_2] \!=\! K_5[Y_2][D_x], [D_xX^{*}] \!=\! K_6[X^{*}][D_x].\) Further considering the conservation laws: \([D_x]+[D_xX^{*}]+[D_xY_2]\,=\,[D_x^T]\) and \([D_y]+[D_yX^{*}]\,=\,[D_y^T]\), one derives:

$$\begin{aligned}{}[D_x]\,&= \,\frac{[D_x^T]}{1+K_4K_5[Y]^2+K_1K_2K_6[S][X]^2},\end{aligned}$$
(8)
$$\begin{aligned}{}[D_xX^{*}]\,&= \,\frac{K_1K_2K_6[S][X]^2[D_x^T]}{1+K_4K_5[Y]^2+K_1K_2K_6[S][X]^2},\end{aligned}$$
(9)
$$\begin{aligned}{}[D_y]\,&= \,\frac{[D_y^T]}{1+K_1K_2K_3[S][X]^2}. \end{aligned}$$
(10)

From the slow reactions, one has:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\hbox {d}[X]}{\hbox {d}t}\,=\,t_1[D_x]+t_2[D_xX^{*}]-d_X[X],\\ \frac{\hbox {d}[Y]}{\hbox {d}t}\,=\,t_3[D_y]-d_Y[Y]. \\ \end{array}\right. \end{aligned}$$
(11)

Replace \([D_x],[D_xX^{*}],[D_y]\) in Eq. (11) with Eqs. (8)–(10), thus one derives Eq. (1).

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Wang, P., Zhang, Y., Lü, J. et al. Functional characteristics of additional positive feedback in genetic circuits. Nonlinear Dyn 79, 397–408 (2015). https://doi.org/10.1007/s11071-014-1673-x

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