Modelling with differential equations

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RESOLVE workshop Modelling of metabolic systems with differential equations

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  • » Change the starting slide numberTo change the slide number that appears on the first slide in your presentation, do the following:On the Design tab, in the Page Setup group, click Page Setup.In the Page Setup dialog box, in the Number slides from box, enter the number that you want shown on the first slide in your presentation, and then click OK.» Handouts printen op eenzwart-wit printer: kies (toch) voorafdrukken in Color (en niet grayscale)» Powerpoint security warning: references to external pictures have been blockedIf you copy an image from a website with Ctrl+C, the chance is very high, that you involuntarily copy a bit of HTML code which causes that warning message.
  • 4,800 BScstudents (3% international)2,800 MScstudents (18% international)Total ~9000 studentsPrograms12 three-yearBachelor’s programs (BSc)28 two-yearMaster’sdegree programs  (MSc)Staff 3,200 employees (30% international) 2,000 academic staffDe TU/e has three Strategic Areas: Energy Health Smart Mobility
  • Bachelor:2 tracks
  • Understanding disease pathways / networksPersonalized Healthcare / Medicinebiomarkerspatient specific interventionguide drug discovery
  • Detailed kinetic models, acute response to metabolic changes, such as stress testsParameter (sensitivity) analysisdecay reaction in Matlab
  • http://web.mit.edu/dc/Policy/MIT%20White%20Paper%20on%20Convergence.pdf
  • Open challengestrategies towards the integration of (bottom-up) systems biology models and more descriptive (top-down) bioinformatics modelsHere: bottom-up
  • Mechanistic / mechanism-based modelsIt is important not only that the behavior of a given system is mimicked by model equations, but also that the model equations are physically / biologically reasonable<-> statistical models
  • Recurrence relation (difference equation)
  • Modelling with differential equations

    1. 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. 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. 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. 4. The Computational Biology group Understanding complex dynamic biochemical systems / biomedical engineering PAGE 416-8-2013 Systems biology Molecular modeling
    5. 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. 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. 7. / biomedical engineering PAGE 716-8-2013 Lecture 1: Modelling with differential equations
    8. 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. 9. Convergence of Life Sciences, Physical Sciences and Engineering • The Three Revolutions / biomedical engineering PAGE 916-8-2013
    10. 10. / biomedical engineering PAGE 1016-8-2013 Models and modeling
    11. 11. / biomedical engineering PAGE 1116-8-2013 M.C. Escher Karl Popper (1902 – 1994) George Box (1919 - 2013)
    12. 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. 13. / biomedical engineering PAGE 1316-8-2013 Nomenclature Processinput output
    14. 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. 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. 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. 17. / biomedical engineering PAGE 1716-8-2013 Differential Equations biology physics model model scheme equations
    18. 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. 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. 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. 21. / biomedical engineering PAGE 2116-8-2013 Simulation of biochemical systems
    22. 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. 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. 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. 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. 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. 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. 28. / biomedical engineering PAGE 2816-8-2013 Computer practical 1: Modelling and simulation of pathways
    29. 29. / biomedical engineering PAGE 2916-8-2013
    30. 30. / biomedical engineering PAGE 3016-8-2013

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