1. R OBUSTNESS ANALYSIS AND TUNING OF SYNTHETIC GENE NETWORKS Grégory Batt, Boyan Yordanov, Ron Weiss, and Calin Belta 1 VC Lab, Dept. of Computer Science, NTHU, Taiwan

2. O VERVIEW Introduction Problems of biological system design Previous solution Problem Piecewise-Multiaffine Models Linear Temporal Logic(LTL) Two problems Conclusion Reference 2 VC Lab, Dept. of Computer Science, NTHU, Taiwan

3. I NTRODUCTION Problems of biological system design Lack of precise knowledge Molecular concentrations Parameter values Previous solution Coarse-grained models Nonlinear differential equation models Stochastic models 3 VC Lab, Dept. of Computer Science, NTHU, Taiwan

4. P IECEWISE -M ULTIAFFINE M ODELS Non-linear model Smooth sigmoidal function 4 VC Lab, Dept. of Computer Science, NTHU, Taiwan

5. P IECEWISE -M ULTIAFFINE M ODELS Non-linear model Smooth sigmoidal function 5 VC Lab, Dept. of Computer Science, NTHU, Taiwan

6. LTL 6 VC Lab, Dept. of Computer Science, NTHU, Taiwan Linear Temporal Logic(LTL) logical operators negation (¬) logical and (^), logical or ( ˇ ) implication () temporal operators future ( F ), globally (G), and until (U) EX:

7. P ROBLEM Let Σ be a PMA system, P an hyperrectangular parameter space, and φ an LTL formula. Robustness: Check whether P is valid for φ. Cant be solve by numerical integration Tuning: Find a set P P such that P is valid for φ. 7 VC Lab, Dept. of Computer Science, NTHU, Taiwan

8. P ROBLEM 1: R OBUSTNESS Discrete abstractions Partition of state space Discrete transition system Finite Test Σ satisfies φ 8 VC Lab, Dept. of Computer Science, NTHU, Taiwan

9. P ROBLEM 2: T UNING 9 VC Lab, Dept. of Computer Science, NTHU, Taiwan

10. P ROBLEM 2: T UNING 15 valid parameters (1.8%) (total 2.27% from 20,000 parameter) No more than ±20% parameter variations 10 VC Lab, Dept. of Computer Science, NTHU, Taiwan

11. C ONCLUSION RoVerGeNe A PMA model, an LTL specification Generate the temporal space and test the property range Parameter tuning 11 VC Lab, Dept. of Computer Science, NTHU, Taiwan

12. R EFERENCE Batt,G. et al. (2007a) Model checking genetic regulatory networks with parameter uncertainty. In Bemporad,A. et al. (eds). Hybrid Systems: Computation and Control, HSCC07 Belta,C. and Habets,L.C.G.J.M. (2006) Controlling a class of nonlinear systems on rectangles. Trans. Aut. Control, 51, 1749–1759. 12 VC Lab, Dept. of Computer Science, NTHU, Taiwan