Minimal intervening control of biomolecular networks leading to a desired cellular state

A cell phenotype can be represented by an attractor state of the underlying molecular regulatory network, to which other network states eventually converge. Here, the set of states converging to each attractor is called its basin of attraction. A central question is how to drive a particular cell state toward a desired attractor with minimal interventions on the network system. We develop a general control framework of complex Boolean networks to provide an answer to this question by identifying control targets on which one-time temporary perturbation can induce a state transition to the boundary of a desired attractor basin. Examples are shown to illustrate the proposed control framework which is also applicable to other types of complex Boolean networks.


Dependence of mHD on the sizes of a network and a basin
To estimate the variance of average mHD as the size of a network increases, update rules are not considered. We assume that a subset k B of the state space  of a given Since there is no state of mHD QED.

Remark 2.
Using Theorems 2 and 3, we obtained implies that the number of values 1 in s is Remark 4. Using Theorems 1, 2, 3, 4 and 5, we suggest that the normalized average mHD has the following upper bound where the property is presented by a simulation example in Supplementary Fig. 7.
and then we obtain So, we suggest that the average mHD decreases for any fixed network size when the two states in k B are farther apart from each other (see Supplementary Fig. 8).
Therefore, using Remarks 2 and 3 together, we obtain and then we suggest that the average mHDs for desired basins of a fixed relative size does not increase even if the network size increases (see Supplementary Fig. 8).

Extension of BRC to a homogeneous system of finite linear differential equations with constant coefficients
The general solution of any homogeneous system of finite linear ordinary differential equations (ODEs) with constant coefficients can in general be explicitly written in a closed form. Using the closed-form solution, we can identify an exact basin and so apply the concept of BRC to the system. This process is explained in the following steps with an example.
Step 1. Construction of the general solution in a closed form Consider a network of three nodes whose dynamics are modelled by the homogeneous system of three linear ordinary differential equations with constant coefficients where i denotes the imaginary number. Then the general solution   X t is written as follows: for generic constants where a V and b V are real and imaginary parts of 2 V such that For the application of BRC we write the general solution   X t in (1) as follows: cos sin cos sin sin cos 2 cos sin 2sin cos Step 2. Identification of the exact basin of a desired attractor We assume that a desired point attractor state is     since the exact basin of the desired attractor is defined as the set of all initial conditions under which the ODE system is convergent to the desired attractor, where R denotes the set of real numbers.
Step 3. Determination of an undesired state The undesired states are any initial state under which the ODE system does not converge to the desired attractor. Letting an initial condition be an undesired state can be the state Step 4. Identifications of control target sets and boundary states We assume that a given control strategy is to identify the minimum number of control target nodes to be perturbed which drive the undesired state into the exact basin (3). If 1 x is perturbed to 0.5 or 2 x is perturbed to 0.5, then the perturbed state can be contained in the exact basin. Therefore minimum control target sets are   respectively. The right side shows the symmetric structure of (F, G).
called "Deterministic" since 1 x is only one input node to 2 x  and each basin state is identified following the reverse of the state trajectory. A deterministic node is denoted by "D". A "nondeterministic" node means that the node is not deterministic and is denoted by "N". e Transformation nodes and the update rules into (1) symmetric nodes, α with the fixed values, the values of nodes 6 and 10 must be changed to value 0 ("control target nodes"), which is represented on the right side of the boxed words. " Step 3" is to find nodes of unfixed values in the basin of 1 which values in α are denoted by "State A". The mHD of state A to the boundary of the reduced basin of 1 is equal to 2, where the reduced basin of 1 is obtained by removing the nodes 5 to 13 from the basin of 1 and state A can be driven to the boundary by using {(node 14, node 21) = (0,1)} ("control target nodes") or {(node 18, node 21) = (0,1)} ("control target nodes"). " Step 4" is to collect control target nodes, which results in two control target sets of α, {(node 6, node 10, node 14, node 21) = (0,0,0,1)} and {(node 6, node 10, node 18, node 21) = (0,0,0,1)}. Therefore perturbation of at least four nodes (mHD = 4) is needed to drive the undesired state  to the basin of  in BRC. "Final step" is to use a control target set to drive α to the boundary of the basin of , after which the state transition flow diagram is presented in top right.   . The right shows that even if the relative size of a basin is extremely small (0.1953%), the average mHD is less than 3.8. b Average mHD decreases as the network size increases and the relative size of a desired basin is fixed. Success rate for 1,310,720 initial states of which collection is the basin of the undesired attractor in the main text. The meanings of symbols and terms in b are equal to those in a. The difference is that all basin states of the undesired attractor are used in b as initial states instead of 2 33 randomly selected initial states used in a. Since all the initial states are undesired basin states, the ratio of all the initial states converging to the desired attractor is equal to 0 (black circle). When the stop time increases up to the 7 th stop time, states are getting close to the desired attractor (this data is not shown) but do not converge to the desired attractor.

Supplementary
plants species (plant 1, plant 2 and plant 3) and two pollinators species (pollinator 1 and pollinator 2) 1 . Arrows and bar-headed lines in the network represent activation and inhibition, respectively. b Restoration of the desired ecological state. Symbol "attr1" is an undesired attractor due to no species in attr1. Symbol "attr2" is an unrealistic attractor due to its cyclic property 1 . Symbol "attr3" is assumed to be a desired attractor.
The control target species to restore the desired ecological state (attr3) from the absence of all species (attr1) are plant 3 and pollinator 2.
and subordinates 2 . Each number in a box denotes a wasp. The right shows that the dominant network has the structure of feed-forward loop without a feedback loop. For the application of BRC, the Boolean logic OR is used for all the interactions. The state of each node in a desired attractor is assumed to be value 1 and undesired states are all states except basin states of the desired attractor. The yellow nodes denote control target nodes, which are located at the tops of dominant relationships and so can be called "master regulators". There is no master regulator except for the control target nodes. b Role of control target nodes in a dominance interaction network with a feedback loop.
The colony of R. marginata is of size 21. Arrows on the left side denote the dominance relationship between dominants and subordinates 2 . The right shows that the dominant network has the structure of feed-forward loop except for the feedback loop 12171412. For the application of BRC, the OR logic is used for all the interactions. The state of each node in a desired attractor is assumed to be value 1 and undesired states are all states except basin states of the desired attractor. The yellow nodes denote control target nodes, which are located at the tops of dominant relationships and so can be called "master regulators". There is no master regulator except for the control target nodes.