Optimal Dynamics Uncertainty Reduction in Gene Networks in the Presence of Experimental Error Daniel N. Mohsenizadeh † , Roozbeh Dehghannasiri † , and Edward R. Dougherty Abstract—We present an experimental design method for choosing optimal experiments to reduce dynamics uncertainty in dynamical gene networks. The method, takes into account both the modeling objective and the experimental error. I. I NTRODUCTION Analysis of network dynamics involving the evolution of entities over time plays a major role in designing drugs. Gene network models usually suffer from inherent uncertainty that manifests dynamics uncertainty. Recently, a dynamical modeling methodology has been proposed that generates dynamical models by computing the system dynamic trajec- tories [1]. The method incorporates available prior knowl- edge from biological databases. However, due to limited prior knowledge, the developed dynamical models embrace dynamics uncertainty. To reduce dynamics uncertainty, one needs to conduct experiments. But, as these experiments are costly, it is beneficial to utilize an experimental design approach to prioritize experiments and then conduct only those with high ranks [2]. In this paper, we propose an experimental design to reduce dynamics uncertainty based on the concept of mean objective cost of uncertainty (MOCU), which measures the uncertainty as the expected increase of operational cost due to uncertainty [3]. We will also show how the error of experiments can be incorporated in the experimental design in an objective-based manner. II. METHOD A network model M =(N,E) contains a set of nodes N representing entities and a set of edges E representing the interactions/processes. The state of a network at the k-th time step, denoted by x k , is determined by the vector of node values at that time step. Each edge has input/output nodes, and possibly a set of control (activator/inhibitor) nodes. An interaction takes place and subsequently its corresponding node values will be updated if all its input and activator nodes are nonzero and its inhibitor nodes are zero. Generally, there are situations where more than one interaction can happen at the same time based on the node values. If prior knowledge determines which interaction happens, this information can be incorporated in the dynamical modeling by assigning edge priority labels [1]. For instance, suppose (e 1 ,e 2 ) ∈ E can happen at the same time. A priority label e prio 1 >e prio 2 means that only the nodes of e 1 will be updated. Due to insufficient prior knowledge some of the edge priority labels are missing. Authors are with the Department of Electrical and Computer Engineer- ing, Texas A&M University, College Station, TX, 77843 USA. (e-mail: danielmz@tamu.edu, roozbehdn@tamu.edu, edward@ece.tamu.edu.) † indicates equal contribution Thus, given M and an initial condition vector x 0 , different updates of the state vector can be computed. These are called dynamic trajectories and denoted by T M x0 . Each trajectory t ∈ T M x0 is a possible sequence of interactions that can take place. The multiplicity of dynamic trajectories is referred to as dynamics uncertainty. To reduce the dynamics uncertainty, or equivalently the number of trajectories, one needs to determine the unknown priority labels. The goal of an experimental design is to find out which priority label is to be determined first via conducting additional experiments. Let θ =(θ 1 ,...,θ U ) represent U unknown priority labels where θ u is the u-th unknown priority label for the pair of edges (e u,1 ,e u,2 ) ∈ E. The network model M can be regarded as an uncertainty class that contains all M θ , where each M θ corresponds to a specific assignment to θ ∈ Θ. Each M θ has a unique trajectory given x 0 . The probability of each model M θ can be calculated as P M (M θ )=  U i=1 P(θ i ). The main objective from network modeling is to design drugs (interventions). To design interventions, first we need to define a criterion for evaluating the behavior of the network from a translational perspective. Let N d ⊂ N be a subset of nodes characterizing a particular phenotype. If v d denotes the vector of desired final/steady-state values for N d and x d,t f represents the final/steady-state vector reached through trajectory t and corresponding to the nodes in N d , then the error ǫ(t) for trajectory t is defined as ǫ(t) := ‖x d,t f − v d ‖. Let Ψ= {ψ j ,j ∈ J } be a class of interventions, where intervention ψ j involves blocking interaction e j ∈ E from happening. Let t M θ x0 (ψ j ) denote the single trajectory of model M θ given x 0 if ψ j is applied. We define the performance error of M θ subsequent to ψ j as E M θ (ψ j )=  x0∈X0 P X0 (x 0 )ǫ ( t M θ x0 (ψ j ) ) , (1) where P X0 (x 0 ) is the probability distribution of the ini- tial conditions. The optimal intervention for model M θ is ψ opt (M θ ) := arg min ψj ,j∈J E M θ (ψ j ). In the presence of uncer- tainty, the average error for intervention ψ j across M is E M (ψ j )=  θ∈Θ  x0∈X0 P M (M θ )P X0 (x 0 )ǫ ( t M θ x0 (ψ j ) ) . (2) The intervention ¯ ψ(M ) = arg min ψj ,j∈J E M (ψ j ) is called robust intervention for M . Consider a set of experiments Ξ= {ξ u ,u =1,...,U }, where experiment ξ u is conducted to find θ u . Given M , P X0 , Ψ, and Ξ, it is of interest to find out which experiment should