Fig. 7

Illustration of comparing different control methods with an example network. a An example network model with a phenotype node P. b Red (white) denotes the value of 1 (0) for each node. c Three categories of control methods where ‘one-to-one’ denotes one initial state to one final state, ‘any-to-one’ denotes any initial state to one desired attractor, and ‘any-to-multiple’ denotes any initial state to one of multiple attractors corresponding to a particular phenotype of interest. d Illustration of the three categories of control methods upon their state spaces. We denote the original state space and the controlled state space as ‘state space (before control)’ and ‘state space (after control)’, respectively. Here, the controlled state space means the state space of the network to which a control set is applied. In the top state space, the original state space contains two states: the left one is an initial state A1=(1, 0, 1, 0, 1, 0, 1) at time t = 0 and the right one is the desired final state B1=(0, 1, 0, 1, 0, 1, 1) at a given time t = T. In this case, the final state B1 is not assumed to be an attractor. The initial state A1 is driven to the final state B1 at t = T in the controlled state space. In the middle state space, the original state space contains two attractors: the left one is an undesired attractor (1, 0, 1, 1, 1, 1, 1) and the right one is the desired attractor (0, 0, 0, 0, 0, 0, 0) whose basin is denoted by dark gray. Here, the basin means a set of states converging to the attractor state. In this case, any initial state is driven to the desired attractor (1, 1, 1, 0, 0, 0, 1) in the controlled state space. In the bottom state space, the desired phenotype value is P = 0. The original state space contains two attractors, (1, 0, 1, 1, 1, 1, 1) and (0, 0, 0, 0, 0, 0, 0), where the second one can be a desired attractor due to P = 0 and its basin is denoted by dark gray. The controlled state space obtained after applying the control set {C = 0} shows that any initial state can be driven to the attractor (0, 0, 0, 0, 0, 0, 0) which has the desired phenotype value P = 0. On the other hand, using the control set {B = 1} instead of {C = 0}, any initial state in the control state space converges to a different attractor (0, 1, 0, 0, 0, 0, 0) of the same desired phenotype value P = 0. e The red dotted links in the top network denote elements of the maximum matching [8], where the node F marked with a red circle indicates a node that is not an end node of any red dotted link and therefore is a unique driver node. In the middle network, the red dotted links denote input links to the nodes C and D marked with red circles, which are elements of mFVS [11, 12]. The bottom network shows the converging tree composed of all control sets that are found based on the Boolean update rules in Fig. 1, where PCK consists of 6 control sets. The process of finding out the control sets is explained in the Result section