1. Identifiability of biological systems Afonso Guerra Assunção Senra Paula Freire Susana Barbosa

2. Identifiability of a biological system A biological system is a group of biomolecules that together perform a certain task. High throughput technologies generates great amount of data  unknown underlying systems. Data quality  observed with noise A system can be described by a set of mass action equations. For simplicity: 0 th, 1 st, 2 nd order

3. Identification problem Equation structure search (model the system- set of ODEs) Experimental design (inputs - initial concentrations) Parameter fitting (optimization problem- minimum least squares) Statistical analysis

4. Equation structure A B C E D F A B A B A B A B A B A B A B Ilustrating:

5. Hypergraph Equation system represented by a directed hypergraph Each edge can involve more than two nodes Nodes  Molecules Edges  Reactions Real world restrictions apply

6. Project Plan 1.Generate GMA for a set of components 2.Choose systems of increasing complexity and simulate dynamic trajectories 3.Parameter inference –Perfect observations and complete knowledge of equation structure –No knowledge of equation structure –With noise: observations and time –Partial knowledge of the equation system 4.Apply evolutionary model

7. Experimental Design Simulation of dinamic trajectories –A priori estimates for parameters –Initial conditions –Algorithm to numerically solve ODEs time Conc

8. Curve fitting Sampling : CHALLENGE in instationary systems –Simultaneous sampling –Sequential sampling –Parameterized sampling Equidistant Exponential Parameter fitting –Widely applied: local search algorithm  requires a good guess! –Estimate parameters one by one  each new step use the previous estimate  1-dimensional problem

9. Curve fitting 1. No noise and complete knowledge Mean least squares distance (MLS) 2. No noise and No knowledge of equation structure Equation Structure Search - Greedy recursive algorithm i.Zeros on the rigth side of the equation set ii.Add a term of the form k x,y [X][Y] or k x [X] iii.Predefine a set of realistic reactions (0.001 < f < 0.05) n components  (n+1)n 2 /2+n 2 possibilities

10. Evolutionary Model Evolution of one model into another Two models: A and B –Full knowledge of B –P A =P B + X,X ~ N (0,t) –Stochastic process: Brownian Motion

11. Questions Stiff equations Nonlinear systems of ODEs – no global optimization guaranteed- stuck in local minima How sufficient and accurate is the data? How reliable are the models? What are the computational challenges?

12. Thank you for your attention! Open Discussion