1. Abstract ODE System Model of GRNs Summary Evolving Small GRNs with a Top-Down Approach Javier Garcia-Bernardo* and Margaret J. Eppstein Department of Computer Science and Vermont Complex Systems Center University of Vermont * jgarciab@uvm.edu Being able to design genetic regulatory networks (GRNs) to achieve a desired cellular function is one of the main goals of synthetic biology 1. While several researchers have used evolutionary algorithms to evolve network motifs or synthetic networks with a desired behavior 2,3, previous studies have rarely evolved minimal or near-minimal GRNs. Even when methods were explicitly designed to try to prune excess parts 4,5, the number of interactions in the evolving GRNs increased over generations. In this paper, we use a top-down approach, where we start with relatively dense GRNs and use differential evolution 6 (DE) to evolve interaction coefficients. When the target dynamical behavior is found embedded in a dense GRN, we narrow the focus of the search and attempt to remove excess interactions. Three methods were tested to encourage parsimonious networks. We first show that the method can quickly rediscover known small GRNs for a toggle switch and an oscillatory circuit. Next we include these known small GRNs as non-evolvable subnetworks in the subsequent evolution of more complex, modular GRNs. By incorporating aggressive pruning after the desired behavior was found, along with a dynamic penalty term, the DE was able to evolve minimal, or nearly minimal, GRNs in all 4 test problems. ) Hierarchical Evolution of GRNs We first validated the method by evolving GRNs for two known circuits: (a) Bistable, a simple toggle switch, and (b) Oscillator, an oscillatory circuit. We then included those known GRNs as non-evolvable subnetworks in the subsequent evolution of two more complex, modular GRNs: (c) ConditionalOscillator, in which we evolved indirect interactions between non-evolvable subnetworks for a toggle switch and an oscillatory circuit, and (d) DualOscillator, which combined two mutually exclusive oscillators; to make this problem more difficult we only included one copy of a non-evolvable oscillatory circuit. Results (out of 25 repetitions each) In summary, we present a top-down, evolutionary approach that can be recursively applied to hierarchically evolve increasingly complex, modular GRNs with few, if any, excess interactions. While our primary motivation is for designing GRNs for synthetic biology, the algorithm introduced in this work could be used for the inference of naturally-occurring GRNs from gene expression data. Additionally, our approach could be used for evolving parsimonious topologies for Artificial Neural Networks. Method Dynamic penalty for # and strength of interactions: Discard weak interactions: if |n i,j | < 0.5, set |n i,j |= 0. Aggressive Pruning (see below) ForcedReductionyesnoyes NoPenaltynoyesno WithPenaltyyes no Aggressive Pruning Encouraging Evolution of Minimal GRNs DE was used to evolve the parameters K i,j and n j,i of up to 6 genes (up to 84 decision variables). Once a desired behavior was found, the DE strategy was shifted from DE/rand/1/bin to DE/rand-to-best/1/bin, to focus the search. DE was combined with three approaches designed to encourage parsimonious networks, as follows: The ForcedReduction method is able to relatively quickly evolve minimal or near-minimal genetic networks with the desired behaviors. The other two methods also succeed in evolving the desired behaviors, but the resulting GRNs have many excess interactions. References In aggressive pruning, at the end of each DE generation subsequent to the strategy shift, for each solution that exhibited the desired dynamics, we iterated through each Hill coefficient n j,i, testing raw fitness (before penalty) with n j,i = 0. If fitness degraded by 10% or less, we kept n j,i = 0. This process was continued until we had replaced five coefficients with zero or we ran out of Hill coefficients to try. ForcedReduction Penalty NoPenalty 1 A.S. Khalil and J.J. Collins, “Synthetic Biology: applications come of age”, Nature Reviews Genetics, 5:367-379, 2010. 2 P. François and V. Hakim, “Design of genetic networks with specified functions by evolution in silico”, PNAS, 101:580–585, 2004. 3 N. Noman, L. Palafox, and H. Iba, “Evolving Genetic Networks for Synthetic Biology”, New Gener. Comput., vol. 31, no. 2, pp. 71–88, 2013. 4 B. Drennan and R. Beer, “Evolution of repressilators using a biologically-motivated model of gene expression”, ALIFE X, pp. 22–27, 2006. 5 M. Dorp, B. Lannoo, and E. Carlon, “Evolutionary Generation of Small Oscillating Genetic Networks”, in Adaptive and Natural Computing Algorithms, pp. 120–129, 2013. 6 R. Storn and K. Price, “Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces”, J. Glob. Optim., 11:341–359, 1997. Small GRNs evolved by the ForcedReduction method. Representative run for the evolution of the DualOscillatory network. For each pair of genes i and j, the coefficients K i,j and n j,i determine the strength of the positive or negative regulatory interaction of the protein coded for by gene j on the expression of gene i. Positive n j,i indicates a negative regulation of the gene i by the protein j, and vice versa; n j,i = 0 means no interaction exists. DNA gene i bound with regulatory proteins p j mRNA m i transcription translation Protein p i regulation Remove 1 interaction if fitness not degraded > 10% for up to 5 interactions removed each generation Initially dense networks Final networks very sparse In ForcedReduction, the number of interactions grows until the desired behavior is found, then excess interactions are removed. Despite more fitness evaluations per generation, this method converges with fewer total fitness evaluations.