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A geometrical approach to control and controllability of nonlinear dynamical networks.

Wang LZ, Su RQ, Huang ZG, Wang X, Wang WX, Grebogi C, Lai YC - Nat Commun (2016)

Bottom Line: To make our framework practically meaningful, we consider restricted parameter perturbation by imposing two constraints: it must be experimentally realizable and applied only temporarily.We introduce the concept of attractor network, which allows us to formulate a quantifiable controllability framework for nonlinear dynamical networks: a network is more controllable if the attractor network is more strongly connected.We test our control framework using examples from various models of experimental gene regulatory networks and demonstrate the beneficial role of noise in facilitating control.

View Article: PubMed Central - PubMed

Affiliation: School of Electrical, Computer and Energy Engineering, Arizona State University, 650 E. Tyler Mall, Tempe, Arizona 85287-5706, USA.

ABSTRACT
In spite of the recent interest and advances in linear controllability of complex networks, controlling nonlinear network dynamics remains an outstanding problem. Here we develop an experimentally feasible control framework for nonlinear dynamical networks that exhibit multistability. The control objective is to apply parameter perturbation to drive the system from one attractor to another, assuming that the former is undesired and the latter is desired. To make our framework practically meaningful, we consider restricted parameter perturbation by imposing two constraints: it must be experimentally realizable and applied only temporarily. We introduce the concept of attractor network, which allows us to formulate a quantifiable controllability framework for nonlinear dynamical networks: a network is more controllable if the attractor network is more strongly connected. We test our control framework using examples from various models of experimental gene regulatory networks and demonstrate the beneficial role of noise in facilitating control.

No MeSH data available.


T-cell survival signalling network and its attractor network.(a) Structure of T-cell survival network: each node is labelled with its generic name, and the arrowhead and diamond-head edges represent activation and inhibition regulations, respectively. The inhibitory edges from ‘Apoptosis' to other nodes are not shown (for clarity). (b) Attractor network of the T-cell network, which contains three nodes: two cancerous states denoted as C1 and C2 and a normal state denoted as N. The two directed edges in the attractor network are multiple, each containing altogether 48 individual edges corresponding to controlling the 48 edges in the original network, which are indicated by the dark dashed lines, whereas the remaining edges in the original network are signified by the light solid lines. Our detailed computations reveal that parameter perturbation on any one of the 48 edges can drive the system from a cancerous state to the normal state. That is, regardless of whether the initial state is C1 or C2, with a proper modification to one of the 48 parameters, the system can be driven to the normal state N.
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f1: T-cell survival signalling network and its attractor network.(a) Structure of T-cell survival network: each node is labelled with its generic name, and the arrowhead and diamond-head edges represent activation and inhibition regulations, respectively. The inhibitory edges from ‘Apoptosis' to other nodes are not shown (for clarity). (b) Attractor network of the T-cell network, which contains three nodes: two cancerous states denoted as C1 and C2 and a normal state denoted as N. The two directed edges in the attractor network are multiple, each containing altogether 48 individual edges corresponding to controlling the 48 edges in the original network, which are indicated by the dark dashed lines, whereas the remaining edges in the original network are signified by the light solid lines. Our detailed computations reveal that parameter perturbation on any one of the 48 edges can drive the system from a cancerous state to the normal state. That is, regardless of whether the initial state is C1 or C2, with a proper modification to one of the 48 parameters, the system can be driven to the normal state N.

Mentions: To demonstrate the construction of attractor networks, we take as an example a realistic biological network, T cells in large granular lymphocyte leukaemia associated with blood cancer. Specifically, apoptosis signalling of the T cells can be described by a network model: T-cell survival signalling network3435, which has 60 nodes and 195 regulatory edges, as shown in Fig. 1a. Nodes in the network represent proteins and transcripts, and the edges correspond to either activation or inhibitory regulations. Experimentally, it was found that there are three attractors for this biophysically detailed network3435. Among the three attractors, two correspond to two distinct cancerous states (denoted as C1 and C2) and one is associated with the normal state (denoted as N). The two cancerous states are biologically equivalent, differing only on node P2 (either activated or inactivated). As the T cells in large granular lymphocyte leukemia disease originate from the failure of the programmed T cells, the normal state corresponds to the situation where the node Apoptosis is activated while all other nodes are inactivated. By translating the Boolean rules into a continuous form using the method described in refs 36, 37 and setting the strength of each edge to unity, one can obtain a set of nonlinear differential equations for the entire network system. Direct simulation of the model indicates that there are three stable fixed point attractors, in agreement with the experimental observation3435. The attractor network is thus quite simple: it has three nodes only, as shown in Fig. 1b. Testing all the 195 experimentally adjustable parameters, we find 48 edges from each cancerous attractor to the normal one (see Supplementary Table 1 for details). Our detailed computations reveal that parameter perturbation on any one of the 48 edges can drive the system from a cancerous state to the normal state. That is, regardless of whether the initial state is C1 or C2, with a proper modification to one of the 48 parameters, the system can be driven to the normal state N. We note that parameter perturbation does exist to drive the system from the normal state to a cancerous state (see Supplementary Note 1 for details).


A geometrical approach to control and controllability of nonlinear dynamical networks.

Wang LZ, Su RQ, Huang ZG, Wang X, Wang WX, Grebogi C, Lai YC - Nat Commun (2016)

T-cell survival signalling network and its attractor network.(a) Structure of T-cell survival network: each node is labelled with its generic name, and the arrowhead and diamond-head edges represent activation and inhibition regulations, respectively. The inhibitory edges from ‘Apoptosis' to other nodes are not shown (for clarity). (b) Attractor network of the T-cell network, which contains three nodes: two cancerous states denoted as C1 and C2 and a normal state denoted as N. The two directed edges in the attractor network are multiple, each containing altogether 48 individual edges corresponding to controlling the 48 edges in the original network, which are indicated by the dark dashed lines, whereas the remaining edges in the original network are signified by the light solid lines. Our detailed computations reveal that parameter perturbation on any one of the 48 edges can drive the system from a cancerous state to the normal state. That is, regardless of whether the initial state is C1 or C2, with a proper modification to one of the 48 parameters, the system can be driven to the normal state N.
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f1: T-cell survival signalling network and its attractor network.(a) Structure of T-cell survival network: each node is labelled with its generic name, and the arrowhead and diamond-head edges represent activation and inhibition regulations, respectively. The inhibitory edges from ‘Apoptosis' to other nodes are not shown (for clarity). (b) Attractor network of the T-cell network, which contains three nodes: two cancerous states denoted as C1 and C2 and a normal state denoted as N. The two directed edges in the attractor network are multiple, each containing altogether 48 individual edges corresponding to controlling the 48 edges in the original network, which are indicated by the dark dashed lines, whereas the remaining edges in the original network are signified by the light solid lines. Our detailed computations reveal that parameter perturbation on any one of the 48 edges can drive the system from a cancerous state to the normal state. That is, regardless of whether the initial state is C1 or C2, with a proper modification to one of the 48 parameters, the system can be driven to the normal state N.
Mentions: To demonstrate the construction of attractor networks, we take as an example a realistic biological network, T cells in large granular lymphocyte leukaemia associated with blood cancer. Specifically, apoptosis signalling of the T cells can be described by a network model: T-cell survival signalling network3435, which has 60 nodes and 195 regulatory edges, as shown in Fig. 1a. Nodes in the network represent proteins and transcripts, and the edges correspond to either activation or inhibitory regulations. Experimentally, it was found that there are three attractors for this biophysically detailed network3435. Among the three attractors, two correspond to two distinct cancerous states (denoted as C1 and C2) and one is associated with the normal state (denoted as N). The two cancerous states are biologically equivalent, differing only on node P2 (either activated or inactivated). As the T cells in large granular lymphocyte leukemia disease originate from the failure of the programmed T cells, the normal state corresponds to the situation where the node Apoptosis is activated while all other nodes are inactivated. By translating the Boolean rules into a continuous form using the method described in refs 36, 37 and setting the strength of each edge to unity, one can obtain a set of nonlinear differential equations for the entire network system. Direct simulation of the model indicates that there are three stable fixed point attractors, in agreement with the experimental observation3435. The attractor network is thus quite simple: it has three nodes only, as shown in Fig. 1b. Testing all the 195 experimentally adjustable parameters, we find 48 edges from each cancerous attractor to the normal one (see Supplementary Table 1 for details). Our detailed computations reveal that parameter perturbation on any one of the 48 edges can drive the system from a cancerous state to the normal state. That is, regardless of whether the initial state is C1 or C2, with a proper modification to one of the 48 parameters, the system can be driven to the normal state N. We note that parameter perturbation does exist to drive the system from the normal state to a cancerous state (see Supplementary Note 1 for details).

Bottom Line: To make our framework practically meaningful, we consider restricted parameter perturbation by imposing two constraints: it must be experimentally realizable and applied only temporarily.We introduce the concept of attractor network, which allows us to formulate a quantifiable controllability framework for nonlinear dynamical networks: a network is more controllable if the attractor network is more strongly connected.We test our control framework using examples from various models of experimental gene regulatory networks and demonstrate the beneficial role of noise in facilitating control.

View Article: PubMed Central - PubMed

Affiliation: School of Electrical, Computer and Energy Engineering, Arizona State University, 650 E. Tyler Mall, Tempe, Arizona 85287-5706, USA.

ABSTRACT
In spite of the recent interest and advances in linear controllability of complex networks, controlling nonlinear network dynamics remains an outstanding problem. Here we develop an experimentally feasible control framework for nonlinear dynamical networks that exhibit multistability. The control objective is to apply parameter perturbation to drive the system from one attractor to another, assuming that the former is undesired and the latter is desired. To make our framework practically meaningful, we consider restricted parameter perturbation by imposing two constraints: it must be experimentally realizable and applied only temporarily. We introduce the concept of attractor network, which allows us to formulate a quantifiable controllability framework for nonlinear dynamical networks: a network is more controllable if the attractor network is more strongly connected. We test our control framework using examples from various models of experimental gene regulatory networks and demonstrate the beneficial role of noise in facilitating control.

No MeSH data available.