1. ALGEBRAIC AND COMBINATORIAL APPROACHES IN SYSTEMS BIOLOGY, MAY 24, 2015 COMBINATORIAL OPTIMIZATION METHODS FOR AGENT-BASED MODELING Matthew Oremland Mathematical Biosciences Institute Ohio State University oremland.2@osu.edu

2. OVERVIEW ABMs in biology Toy model for demonstration Conversion to equation system Pareto optimization as evolutionary algorithm Three dimensional visualization Potential applications to ACSB talks

3. AGENT-BASED MODELS IN BIOLOGY An, G. In-Silico Experiments of Existing and Hypothetical Cytokine-Directed Clinical Trials using Agent Based Modeling. Critical Care Medicine 2004; 32(10):2050-2060. Systematic Inflammatory Response Syndrome

4. AGENT-BASED MODELS IN BIOLOGY Cell dynamics in colon crypts Bravo, R. and D.E. Axelrod. (2013)A calibrated agent-based computer model of stochastic cell dynamics in normal human colon crypts useful for in silico experiments. Theoretical Biology and Medical Modeling 10:66. DOI: 10.1186/1742-4682-10-66.

5. ADVANTAGES AND ISSUES Advantages: Easy to capture heterogeneity Adaptable to a wide range of behaviors Local interplay well-suited for systems approach Visualization as a interdisciplinary middle ground Issues: Many parameters Control theory and optimization methods not well-developed

6. FRAMEWORK OVERVIEW

7. PREDATOR-PREY: A RUNNING EXAMPLE Optimization problem: On each of 100 days of simulated time, we can choose to poison take rabbits to a day spa or not. What is the best 100-day schedule if our goal is to minimize the total number of rabbits throughout the run while also minimizing the total day spa budget amount of poison?

8. DISCLAIMER There are some individuals who have no objection to killing rabbits.

9. PREDATOR-PREY: A RUNNING EXAMPLE Time-discrete difference equations capture the number of rabbits and the level of grass at each time step. Parameters a, b, and c are determined using nonlinear least-squares regression. a = 0.8445, b = 0.3743, c = 1.0043

10. PREDATOR-PREY: A RUNNING EXAMPLE We introduce control into the equations. Here, u(t) is a vector consisting of 100 entries: 0 := poison is not used 1 := poison is used

11. EQUATION VS. ABM DYNAMICS

12. OPTIMIZING OVER 2 100 CONTROLS Goal: Determine u(t) so that we minimize r(0) + r(1) + … + r(100) and u(0) + u(1) + … u(100). Begin with population of random controls U For each, plot and

13. OPTIMIZING OVER 2 100 CONTROLS Determine the Pareto frontier Not Pareto optimalPareto optimal

14. OPTIMIZING OVER 2 100 CONTROLS Niche count = nearby neighbors Select parents for crossover based on Pareto optimality and low niche count Re-combine parents to generate next generation Add mutation to encourage exploration Repeat until convergence

15. INITIAL AND FINAL GENERATIONS Selection of optimal control is a ‘managerial’ decision

16. EXTENDING TO THREE OBJECTIVES f 1 = (x 1 – 2) 2 + (x 2 -3) 2 f 2 = (x 1 – 2) 2 + (x 2 -1) 2 + 1 f 3 = (x 1 – 3) 2 + (x 2 -2) 2 + 2 Minimize: subject to x 1, x 2 in [-4,4]

17. EXTENDING TO THREE OBJECTIVES

19. POTENTIAL RELEVANT APPLICATIONS Stillman: edge number vs. power-law distribution fit Murrugarra: edge vs. node deletion cost Vera-Licona: priority of major and minor objectives He: Parameter weights Hao: Minimize partitions and error Sturmfels: Maximize desired entry, minimize sequence length, minimize undesirable entry

20. SUMMARY Pareto optimization allows for global optimization without need of a cost function Provides a suite of solutions Appropriate for situations in which enumeration is not feasible Very few conditions need to be met in order to use it Can be applied directly to simulation models without need of equation approximation