Parameter synthesis for biological systems modelling Thao Dang, Eric - PowerPoint PPT Presentation

Parameter synthesis for biological systems modelling Thao Dang, Eric Fanchon, Marcelo Forets, Victor Magron, Alexandre Rocca VERIMAG/CNRS and TIMC-IMAG, Grenoble Universit´ e Grenoble Alpes, France France SynCop and PV 14-15 April 2018, Thessaloniki, Greece Parameter synthesis for biological modelling 1 / 27

Biological Modelling Parameter synthesis for biological modelling 2 / 27

Biological Modelling Parameter synthesis for biological modelling 2 / 27

Biological Modelling Evolution over time of certain quantities (e.g. concentrations of species) From pathway structure, equations are established using kinetic laws (e.g. mass action or Michaelis-Menten law) Parameter synthesis for biological modelling 3 / 27

Parameters in Biological Models and Formal Methods Parameters used to capture uncertainty in extrapolation of experimental data variety in species Formal Methods efficiently handling uncertainty tubes of behaviors rather than single behavior obtained by numerical simulation Parameter synthesis for biological modelling 4 / 27

Parameter Estimation ➢ Only some parameters (such as reaction rates, and production and decay coefficients) have phical meanings and can be experimentally measured ➢ Others come from approximations or reductions, therefore might have no direct biological or biochemical interpretation. ➢ Fitting : using time series of experimental measurements to determine parameter values that minimize error between these measurements and corresponding model predictions. ➢ Measurements can be very noisy, and obtained at a limited number of time points. Parameter synthesis for biological modelling 5 / 27

Model Revision We have : ➢ A First ODE model (of Haemoglobin production) ➢ Data from multi-stage experimental procedures. ➢ Previous results from simulations on a Monte-Carlo sampling of the parameter space Parameter synthesis for biological modelling 6 / 27

Model Revision We have : ➢ A First ODE model (of Haemoglobin production) ➢ Data from multi-stage experimental procedures. ➢ Previous results from simulations on a Monte-Carlo sampling of the parameter space Parameter synthesis for biological modelling 6 / 27

Model Revision We have : ➢ A First ODE model (of Haemoglobin production) ➢ Data from multi-stage experimental procedures. ➢ Previous results from simulations on a Monte-Carlo sampling of the parameter space Parameter synthesis for biological modelling 6 / 27

Model Revision We have : ➢ A First ODE model (of Haemoglobin production) ➢ Data from multi-stage experimental procedures. ➢ Previous results from simulations on a Monte-Carlo sampling of the parameter space We want to : Parameter synthesis for biological modelling 6 / 27

Model Revision We have : ➢ A First ODE model (of Haemoglobin production) ➢ Data from multi-stage experimental procedures. ➢ Previous results from simulations on a Monte-Carlo sampling of the parameter space We want to : ☞ Improve the quality of the fitting Parameter synthesis for biological modelling 6 / 27

Model Revision We have : ➢ A First ODE model (of Haemoglobin production) ➢ Data from multi-stage experimental procedures. ➢ Previous results from simulations on a Monte-Carlo sampling of the parameter space We want to : ☞ Improve the quality of the fitting How ? Parameter synthesis for biological modelling 6 / 27

Model Revision We have : ➢ A First ODE model (of Haemoglobin production) ➢ Data from multi-stage experimental procedures. ➢ Previous results from simulations on a Monte-Carlo sampling of the parameter space We want to : ☞ Improve the quality of the fitting One solution ➢ Transform a parameters k into a time varying function k ( t ) Parameter synthesis for biological modelling 6 / 27

Model of Haemoglobin Production ➢ Model of Haemoglobin production, from ANR Cadmidia Project ➢ 52 first hours of an eryhtroblast, before leaving the bone marrow Parameter synthesis for biological modelling 7 / 27

Experimental Workflow Parameter synthesis for biological modelling 8 / 27

Model of Hemoglobin Production Mechanism F norm : System without radioactive species. dFe  = k 1 Fe ext − k 2 Fe − k 3 ( t ) Fe dt  dH  = k 3 ( t ) Fe − k 4 H − 4 k 5 HG  dt dG = k 6 H − 4 k 5 HG − k 7 G dt   dHb  = k 5 HG − k 8 Hb dt F rad : System with (additional) radioactive species.  . . . = . . .   d 59 Fe  = k 1 59 Fe ext − k 2 59 Fe − k 3 ( t ) 59 Fe  dt   d 59 H = k 3 ( t ) 59 H − k 4 59 H − 4 k 5 59 HG rad dt d G rad  = k 6 ( 59 H + H ) − 4 k 5 ( 59 H + H ) G rad − k 7 G rad  dt   d 59 Hb  = 59 HG rad − k 8 59 Hb k 5  dt Parameter synthesis for biological modelling 9 / 27

Modelling as an Hybrid Automaton Parameter synthesis for biological modelling 10 / 27

Parameter Synthesis by Optimal Control Approach ➢ Formulating reachable sets by occupation measures ([Henrion et 2013, Shia 2014, Zhao 2017] ➢ Formulating data fitting error as objective function ➢ Solving optimal control problem for hybrid automata to find inputs allows best data fitting Parameter synthesis for biological modelling 11 / 27

Occupation Measures : Basic Definitions ➢ Measure µ defined on a set X ⊂ R n s.t. µ ( ∅ ) = 0 and µ ( � ∞ X i ) = � ∞ 1 µ ( X i ) (pairwise disjoint collection of sets X i ⊂ X ) i ➢ Dynamical system ˙ x ( t ) = f ( t , x , u ) where t in [0 , T ], x ∈ X , u ∈ U Given an initial state x 0 , and a trajectory x ( t | x 0 ) under an input u ( t | x 0 ) ∈ U , occupation measure µ ( ·| x 0 ) for ∆ × X × U ⊂ [0 , T ] × X × U � T µ (∆ × X × U | x 0 ) := I ∆ × X × U ( t , x ( t | x 0 ) , u ( t | x 0 )) dt 0 I is the indicator function This measure defines the time the system spends in × X × U . ➢ Property : for any measurable function g ( t , x , u ) � T � T g ( t , x ( t | x 0 ) , u ( t | x 0 )) dt = I [0 , T ] ×X×U g ( t , x , u ) d µ ( t , x , u | x 0 ) 0 0 Parameter synthesis for biological modelling 12 / 27

Occupation Measures Linear operator for continuously differentiable functions ∂ t + � n v ∈ C 1 ([0 , T ] , X ), v → L v := ∂ v ∂ v ∂ x i f i ( t , x , u ) i =1 Adjoint operator �L ′ v , µ � := � µ, L v � = � [0 , T ] ×X×U L v d µ ( t , x , u | x 0 ) Then, for a test function v ∈ C 1 ([0 , T ] , X ) v ( T , x ( T | x 0 )) = v (0 , x 0 ) + �L ′ µ ( ·| x 0 ) , v � Parameter synthesis for biological modelling 13 / 27

Occupation Measures for Reachability Analysis Now we consider initial measure µ 0 for the state space X . Initial state x 0 ∈ spt ( µ 0 ) Average occupational measure for a set ∆ × X × U ⊂ [0 , T ] × X × U , � µ (∆ × X × U ) = µ (∆ × X × U | x 0 ) d µ 0 ( x 0 ) X � For final time T , µ T ( X ) = X I X ( x ( T | x 0 )) d µ 0 ( x 0 ) Liouville equation (conservation of the information along the trajectory for any test function v ) � � � v ( T , x ) d µ T ( x ) = v (0 , x ) d µ 0 ( x )+ L v ( t , x , u ) d µ ( t , x , u ) X T X [0 , T ] × X × U More concisely, � µ T , v � − � µ 0 , v � = � µ, L v � Parameter synthesis for biological modelling 14 / 27

Occupation Measures for Hybrid Systems Consider a transition from location j to location i , with guard G j , i , X i is the state space of location i . Final occupation at location i : occupation at location i by continuous dynamics and occupation by taking transitions to i µ i � T = µ T ( X i ) + µ ( G j , i ( j , i ) If transition ( j , i ) has a reset map R j , i , then � µ ( R j , i ( G j , i )) , v � = � R ∗ j , i µ ( G j , i ) , v � 0 = µ 0 ( X i ) + R ∗ For initial measure, µ i j , i µ ( G j , i ) Liouville’s equation L ′ µ i = µ T ( X i ) + � µ ( G ( j , i )) − µ 0 ( X i ) − R ∗ j , i µ ( G j , i ) ( j , i ) Parameter synthesis for biological modelling 15 / 27

Optimal Control Problem ➢ Terminal time cost � T inf h ( t , x ( t ) , u ( t )) dt + H ( x ( T )) . ( x , u ) 0 ☞ Additional costs at intermediate time points T j � T � inf h ( t , x ( t ) , u ( t )) dt + H ( x ( T j )) ( x , u ) 0 0 ≤ j ≤ n exp Parameter synthesis for biological modelling 16 / 27

Optimal Control Problem Formulation in Occupation Measure � � µ i h i + µ i inf T H i i ∈ I i ∈ I such that L ′ µ i = µ T ( X i ) + � µ ( G ( j , i )) − µ 0 ( X i ) − R ∗ j , i µ ( G j , i ) ( j , i ) and constraints of positivity of measure variables Parameter synthesis for biological modelling 17 / 27

Finite Dimensional Semidefinite Program Formulation Given a multi-index α ∈ N n , moments of a measure µ is x α d µ ( x ) � [ y µ ] α := � | α |≤ r p α x α ) d µ . Polynomial p ∈ R r [ x ], let L y µ ( p ) := � µ, p � = ( � Moment matrix [ M r ( y µ )] ( α,β ) := [ y µ ] ( α + β ) , where | α + β | ≤ 2 r . Given a polynomial g of degree l < r , localizing matrix [ L r ( g , y µ )] ( α,β ) := � | γ |≤ l g γ [ y µ ] ( α + β ) ⇒ Rewrite the positivity constraints as semidefinite constraints on moments and localizing matrices, and then truncating the degree of the moments to 2 r ⇒ finite dimensional semidefinite program [Nesterov, Lasserre, Parrilo (2000-)] Parameter synthesis for biological modelling 18 / 27

Hybrid automaton of Hemoglobin Production Parameter synthesis for biological modelling 19 / 27