The Multistable Behavior Produced By The Networks Coupled Of C FFL And FBL

The feedforward loop(FFL),composed of three substances X,Y,and Z,is a common motif in biological networks.The FFL contains two effects,one is the direct effect of X on Z,the other is the indirect effect of X on Z through Y.If these two effects are both activation or repression,this kind of FFL is called a cohenrent feedforward loop(c FFL).According to the input function,the c FFL can be classified as "AND" type(both X and Y are needed to activate Z)and "OR" type(either X or Y is sufficient to activate Z).In the network,the regulation that a component regulates its own directly or indirectly is called feedback effect.The loop formed by the components participating in this effect is called feedback loop(FBL).The existence of FBL is a necessary condition for the emergence of complex dynamic behaviors,including multiple steady states.Although FFL and FBL have been extensively studied,respectively,there are many instances that FFL and FBL coexist in biological networks,and there is still a lack of clear understanding on their dynamic behaviors.To study the role of FBL in c FFL producing multistable behaviors,we first enumerate the729 topological structures formed by them and describe these structures by differential equation models.Then we randomly sample parameters in the model and find the number of equilibrium solutions of the equation to obtain the probability of each topological structure producing multistability.Finally,we carry out clustering analysis of the topological structure,and calculate joint and conditional probability between the manners of feedback effects.In addition,the nonlinear dynamic analysis is employed for the topological structure with the highest probability of generating multistability.The results show that for the "AND" c FFL,the positive feedback effect of Z on itself is crucial to improve the probability that system owns multistability;for the "OR" c FFL,the positive feedback effect of X on itself plays a key role in improve multistable probability of systems.We systematically study the multistable behavior of the networks coupled of c FFL and FBL for the first time.This research reveals the critical contributors for the multistability of the network and advances our understanding of the research on the dynamic of motifs in biological networks.