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RESOLVE workshop Modelling of metabolic systems with differential equations

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- 1. RESOLVE workshop May 15, 2013 Natal van Riel, Christian Tiemann, Fianne Sips Eindhoven University of Technology, the Netherlands Dept. of Biomedical Engineering, n.a.w.v.riel@tue.nl Systems Medicine and Metabolic Diseases
- 2. The university • A research university specializing in engineering science & technology • 9 departments • Biomedical Engineering • Built Environment • Electrical Engineering • Industrial Design • Industrial Eng. & Innovation Sciences • Chemical Eng. and Chemistry • Applied Physics • Mechanical Engineering • Mathematics and Computer Science • Students • 4,800 BSc students • 2,800 MSc students • 200 technological designers (PDEng) • 1,100 doctoral candidates (PhD) • Strategic Research Areas • Energy • Health • Smart Mobility / biomedical engineering PAGE 216-8-2013
- 3. The Biomedical Engineering department • 8 groups • Soft tissue biomech. & eng. (Baaijens & Bouten) • Cardiovasculair biomechanics (van de Vosse) • Orthopaedic biomechanics (Ito) • Chemical biology (Brunsveld) • Biomedical chemistry (Meijer) • Biomedical NMR (Nicolaij) • Biomedical image analysis (ter Haar- Romeny) • Computational biology (Hilbers) • Bachelor • Biomedical Engineering • Medical Sciences and Technology (Sept. 2012) • Master • Biomedical Engineering • Medical Engineering • 3 thematic research programs • Regenerative Medicine • Molecular Imaging • Systems Medicine / biomedical engineering PAGE 38/16/2013
- 4. The Computational Biology group Understanding complex dynamic biochemical systems / biomedical engineering PAGE 416-8-2013 Systems biology Molecular modeling
- 5. Program / biomedical engineering PAGE 516-8-2013 Day 1 (Wed. May 15, 2013) • 10:00 Opening • 10:15 Lecture 1: Modelling with differential equations • 11:10 Coffee break • 11:30 Computer practical 1: Modelling and simulation of pathways • 13:00 lunch • 14:00 Lecture 2: Parameter estimation • 15:00 Computer practical 2: Parameter estimation in practice • 16:15 Pitch talks by participants • 16:45 Discussion about possibilities to model data provided by the participants. Select cases that are interesting to be explored in more detail. • 17:30 Drinks • 19:00 Workshop dinner
- 6. Program Day 2 (Thu. May 16, 2013) • 9:00 Inquiry of observations and (remaining) questions of day 1 • 9:15 Lecture 3: Introducing ADAPT • 10:10 Coffee break • 10:30 Computer practical 3: ADAPT • 12:30 lunch • 13:30 Computer practical 4: • Option 1 Work with your own data • Option 2 Continue working on previous practicals • 15:30 Wrap-up discussion • 16:00 Closure / biomedical engineering PAGE 616-8-2013
- 7. / biomedical engineering PAGE 716-8-2013 Lecture 1: Modelling with differential equations
- 8. Contents • Models • Nomenclature • Differential Equations • decay reaction • Simulation • numbers required • decay reaction in Excel and Matlab • Computer practical 1: • modelling and simulation of pathways • irreversible enzymatic reaction / biomedical engineering PAGE 816-8-2013
- 9. Convergence of Life Sciences, Physical Sciences and Engineering • The Three Revolutions / biomedical engineering PAGE 916-8-2013
- 10. / biomedical engineering PAGE 1016-8-2013 Models and modeling
- 11. / biomedical engineering PAGE 1116-8-2013 M.C. Escher Karl Popper (1902 – 1994) George Box (1919 - 2013)
- 12. Mathematical models in systems biology TOP-DOWN • Bioinformatics • ‘omics’-based • Statistics / statistical models • Hypothesis driven • Targeted measurements • Differential equations BOTTOM-UP / biomedical engineering PAGE 1216-8-2013
- 13. / biomedical engineering PAGE 1316-8-2013 Nomenclature Processinput output
- 14. Nomenclature and definitions Dynamic systems with input(s) and output(s) • (State) variables: x, dynamics x(t) • (Independent) input: u, u(t) • Output: y, y(x,u,t) • 0, , ,x u y t model f(x,u,t) u(t) y(t) ENVIRONMENT input • experimental perturbations output process • observations • measurements output model process
- 15. Nomenclature and definitions • Derivative: • Scalar, vector: • Parameters: '( )x t x ( ) , dx t dx dt dt 2 2 , "( ), d x x t x dt , ,x x x 1 2[ , ,..., ]nx x x x 1 2T n x x x x 0 n x , ,p p p
- 16. / biomedical engineering PAGE 1616-8-2013 Nomenclature and definitions • Differential equations • Ordinary Differential Equation (ODE) • Only derivatives w.r.t. one of the independent variables Here: time (dynamics) • Partial DE (PDE) E.g. also derivatives in space • Autonomous • Steady-state: rate of change = 0 • Stable / unstable • Bistable 2 2 2 c t x heat equation 0ˆ dx dt ( ( ), ( ), ) dx f x t u t p dt ( ( ), ) dx f x t p dt
- 17. / biomedical engineering PAGE 1716-8-2013 Differential Equations biology physics model model scheme equations
- 18. Differential Equations (DE) • Mathematical framework to describe a deterministic relation involving continuously varying quantities (modeled by functions) and their rates of change in time and/or space (expressed as derivatives) Back to Sir Isaac Newton (classical mechanics) • Newton's laws allow one to predict the unknown position of a body as a function of time (trajectory) in relation to the position, velocity, acceleration and various forces acting on the body • DE’s can describe real world (physical, chemical, biological) processes (that ‘live’ in continuous time) • DE’s can capture mechanistic understanding • DE’s play a prominent role in many areas of science and technology (engineering, physics, economics, …) / biomedical engineering PAGE 1816-8-2013
- 19. / biomedical engineering PAGE 1916-8-2013 From a ‘wiring diagram’ to a set of ODEs • Mass balance for each species Change in concentration = (producing reactions) - (consuming and degrading processes) • Each mass balance will translate into a (1st order) differential equation • Species are coupled through interactions (biochemical conversions) network system of coupled differential equations ( )dx t dt
- 20. An irreversible monomolecular reaction • An irreversible monomolecular reaction Autonomous system (chemistry: closed system) • Law of mass action • Model with y = [A] • Initial condition • Solution • Required: values for parameter(s) and initial conditions k A dy ky dt (0) 5y1k / biomedical engineering PAGE 2016-8-2013 0 kt y A e 0(0)y A
- 21. / biomedical engineering PAGE 2116-8-2013 Simulation of biochemical systems
- 22. / biomedical engineering PAGE 2216-8-2013 Simulation of 1st order ODE’s Numerical: discretization (here equidistant) • Taylor series expansion • For small td the higher powers td 2, td 3, … are very small. This suggests the crude approximation • Forward Euler method (1st order fixed step method): 2 [ 1] [ ] '[ ] ''[ ] . . . 2 d d t y i y i t y i y i H OT [ 1] [ ] '[ ] [ ] ( )d dy i y i t y i y i t f i [ 1] [ ] ( , [ ])dy i y i t f i y i '( ) ( ( ))y t f y t y(0) = y0 td 2td0 [ ] ( ),dy i y it i
- 23. / biomedical engineering PAGE 2316-8-2013 Forward Euler method • A recurrence relation (Difference Equation) i i+1 slope= f (y[i]) td y[i] y[i+1] [ 1] [ ] ( , [ ])dy i y i t f i y i Example: molecular decay
- 24. / biomedical engineering PAGE 2416-8-2013 Effect of integration step td on accuracy • exact solution • td = 1 • td = 0.1 • td = 0.01 0 2 4 6 8 10 0 1 2 3 4 5 Euler integration k=1 - ( ) (0) 5kt t y t y e e y ky
- 25. / biomedical engineering PAGE 2516-8-2013 Using computers to simulate (bio)chemical kinetics • A great number of computer tools is available for simulation of systems of coupled DE’s • Matlab Python − Systems Biology Tlbx - PySCeS (Python Simulator − for Cellular Systems) • Supply a code that computes the time derivatives of the ‘state variables’ (right-hand side of 1st order differential equations) • Graphical modeling and simulation tools
- 26. / biomedical engineering PAGE 2616-8-2013 1st order fixed step method • 1st order fixed step method • Euler: • In Matlab: • t1, tend, td and x0 depend on the system and the simulation • p is a vector with the model parameters • x is matrix with different time points as the rows and the states in the columns [ 1] [ ] ( )dx i x i t f i tspan=t1:td:tend; x(1,:)=x0; for i=1:length(tspan)-1 x(i+1,:)=x(i,:)+td*f(i,x(i,:),p); end function dx=f(i,x,p) … %enter the ODE’s here ( ) ( ( ))x t f x t x(0) = x0 autonomous system:
- 27. / biomedical engineering PAGE 2716-8-2013 Variable step integration methods • Higher order, variable step method • In Matlab: • ‘options’ defines settings of the simulation algorithm and can be changed using odeset; usually default (options=[]) is OK • all input arguments of ode15s after ‘options’ are user defined; the function with the ODE’s has to accept these as the 3rd (and so forth) inputs • t is determined by Matlab tspan=[t1,tend]; [t,x]=ode15s(@f,tspan,x0,options, p); %see help ode15s function dxdt=f(t,x, p) … %enter the ODE’s here dxdt=dxdt(:); %ode45 requires output to be a column ( ) ( ( ))x t f x t x(0) = x0
- 28. / biomedical engineering PAGE 2816-8-2013 Computer practical 1: Modelling and simulation of pathways
- 29. / biomedical engineering PAGE 2916-8-2013
- 30. / biomedical engineering PAGE 3016-8-2013

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