alia joko 21st Nordic Process Control Workshop 19.01.18 Neural network-based pruning and sensitivity analysis of an apoptosis model alia joko 21st Nordic Process Control Workshop 19.01.18

Apoptosis – what and why? Programmed and morphologically distinct form of cell death Relevance Development Sculpting of fingers Homeostasis Millions of blood cells eliminated daily to balance production Immune Surveillance Killing cancer and virus infected cells

Apoptosis – information processing and response Large number of signalling proteins Complex interactions Emergent Systems behaviour What is the ‘point of no return’ ? How robust is this decision point? Which signalling motifs are responsible – positive feedback loops, competitive inhibition Why are some cells in a population more sensitivity to cell death stimuli than others? What principles underlie this variability?

Mechanistic Apoptosis Model Formalisms Cell 2011 144, 926-939DOI: (10.1016/j.cell.2011.03.002)

Data-driven models of signal transduction New technologies are permitting large-scale quantitative studies of signal transduction networks Data-driven modelling approaches becoming the standard tools Which are the essential features, Nature Cell Biology 8, 1195–1203 (2006) doi:10.1038/ncb1497

Modelling by Feedforward Networks No or little a priori knowledge of the process required Captures nonlinear relations between process variables Works in arbitrary dimensions Resulting models are fast Requires many data points Default network architecture not a good choice Risk of Trash in Trash out May overfit the data Remedies Judicious choice of inputs based on “process” knowledge Limit the architecture (“don’t use too many hidden nodes”) Pruning large networks or let small network grow Simultaneous optimization of weights and connections Judicious choice of inputs based on “process” knowledge Limit the architecture (“don’t use too many hidden nodes”) Pruning large networks or let small network grow Simultaneous optimization of weights and connections

Saxén - Petterson neural network pruning algorithm Multiple networks evolved by a pruning algorithm Rank the resulting models, retaining the best ones Appearance of the inputs in these reflects their importance Pruning Algorithm Randomly generate lower-layer weights, W0. Set i = 1. Reset in turn each weight, j, in Wi and determine the upper layer weights, wj , by linear least squares, giving the objective function value Fij Set Wi = Wi-1 and reset permanently the weight with minimum Fij. Call this index Set and save this variable in a book-keeping matrix Set i = i + 1. If i < n N, go to 2, else end. Fij w W

Bentele-Toivonen Apoptosis Model Single cell model with 41 molecules, 32 reactions, 50 parameters to be estimated Reactions modelled by mass action and Michaelis- Menten kinetics Stochastic log-normal distributions of protein concentrations and reaction rates to mimic cell population Populations with 1000 cells simulated Time to apoptosis :

Results from the network pruning Networks with 10 hidden nodes and 59 inputs were pruned 500 times from different starting weight matrices For each lower layer complexity the run with the minimum error is retained Minimum approximation error

Relevant model parameters The occurrences of 56 inputs on the Pareto fronts of the 20 best models show 11 relevant parameters

Model – data fit

Local sensitivity analysis

Reduced model : new insights? Experimentally verified

Summary Fast and easy generation of promising reduced model subsets (as collected on Pareto fronts) Successfully finds relevant parameters in a complex model of cell death (apoptosis) Provides a means of analysing the sensitivity of the output to the identified relevant parameters Introduces possibilities for experiment design

Henrik Saxén, Prof Frank Petterson, Ph.D Acknowledgements Henrik Saxén, Prof Frank Petterson, Ph.D Thank you for your attention