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Numerical integration

Numerical integration. x. Initial value problem. function dy = GeneDE ( t,y,p ) betam = p(1); alpham = p(2); q = p(3); alpha = p(4); dy = zeros(2,1); dy (1) = betam - alpham *y(1); dy (2) = q*y(1) - alpha*y(2 );. MatLab example.

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Numerical integration

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  1. Numerical integration x Initial value problem functiondy = GeneDE(t,y,p) betam = p(1); alpham = p(2); q = p(3); alpha = p(4); dy = zeros(2,1); dy(1) = betam - alpham*y(1); dy(2) = q*y(1) - alpha*y(2); MatLab example betam= 1/10; alpham= 1/30; q = 10; alpha = 1/30;% Parameters MaxTime= 60; % Integration time Yinitial= 0; Yminitial= 0;% Initial conditions [T,Y] = ode45(@GeneDE,[0 MaxTime],[YminitialYinitial],[],[betamalpham q alpha]); h = figure; subplot(2,1,1); plot(T,Y(:,1),'r','LineWidth',2); set(gca,'FontSize',12); title('Model with mRNA and protein explicitly separated: mRNA'); xlabel('t'); ylabel('Ym'); subplot(2,1,2); plot(T,Y(:,2),'LineWidth',2); set(gca,'FontSize',12); title('Model with mRNA and protein explicitly separated: Protein'); xlabel('t'); ylabel('Y'); t

  2. Initial value problem: Equations x Given: How a system state turns into a future system state Given: Initial condition Want: Trajectory of system x0(t0) t

  3. Initial value problem: Illustration x x0(t0) h t

  4. First approximation: Euler method xEULER(t0 + h) x x0(t0) h t

  5. Back up a bit to estimate more representative slope xEULER(t0 + h) x x0(t0) h t

  6. Back up a bit to estimate more representative slope xEULER(t0 + h) x x2ND ORDER(t0 + h) x0(t0) h t

  7. Iterate . . . x x0(t0) t

  8. Iterate again . . . x x0(t0) t

  9. Iterate yet again . . . x x0(t0) t

  10. Error accumulates in the numerical solution x x(t) x0(t0) x2ND ORDER(t) t

  11. Quality control: Adaptive stepsize x x0(t0) h t

  12. Quality control: Adaptive stepsize x x0(t0) h/2 t

  13. Quality control: Adaptive stepsize x x0(t0) t

  14. Quality control: Adaptive stepsize x x0(t0) t

  15. Numerical integration x Initial value problem functiondy = GeneDE(t,y,p) betam = p(1); alpham = p(2); q = p(3); alpha = p(4); dy = zeros(2,1); dy(1) = betam - alpham*y(1); dy(2) = q*y(1) - alpha*y(2); MatLab example betam= 1/10; alpham= 1/30; q = 10; alpha = 1/30;% Parameters MaxTime= 60; % Integration time Yinitial= 0; Yminitial= 0;% Initial conditions [T,Y] = ode45(@GeneDE,[0 MaxTime],[YminitialYinitial],[],[betamalpham q alpha]); h = figure; subplot(2,1,1); plot(T,Y(:,1),'r','LineWidth',2); set(gca,'FontSize',12); title('Model with mRNA and protein explicitly separated: mRNA'); xlabel('t'); ylabel('Ym'); subplot(2,1,2); plot(T,Y(:,2),'LineWidth',2); set(gca,'FontSize',12); title('Model with mRNA and protein explicitly separated: Protein'); xlabel('t'); ylabel('Y'); t

  16. MatLab example x Given: How a system state turns into a future system state Given: Initial condition x0(t0) Want: Trajectory of system t

  17. MatLab example Script: Tell MatLab initial conditions, parameters, and time intervals. functiondy = GeneDE(t,y,p) Integrate the DE system. [T,Y] = ode45(@GeneDE…) Plot results.

  18. Create a file called GeneDE.m Script: Tell MatLab initial conditions, parameters, and time intervals. functiondy = GeneDE(t,y,p) %Extract parameters from p betam= p(1); alpham = p(2); q = p(3); alpha = p(4); % Format output as a column vector dy= zeros(2,1); % Specify differential equations dy(1) = betam - alpham*y(1); dy(2) = q*y(1) - alpha*y(2); Integrate the DE system. [T,Y] = ode45(@GeneDE…) Plot results.

  19. Create a file called RunGeneDE.m Script: Tell MatLab initial conditions, parameters, and time intervals. functiondy = GeneDE(t,y,p) betam = p(1); alpham = p(2); q = p(3); alpha = p(4); dy = zeros(2,1); dy(1) = betam - alpham*y(1); dy(2) = q*y(1) - alpha*y(2); Integrate the DE system. [T,Y] = ode45(@GeneDE…) Plot results.

  20. Fill in RunGeneDE.m and run betam= 1/10; alpham= 1/30; q = 10; alpha = 1/30;% Parameters MaxTime= 60; % Integration time Yinitial= 0; Yminitial= 0;% Initial conditions [T,Y] = ode45(@GeneDE,[0 MaxTime],[YminitialYinitial],[],[betamalpham q alpha]); h = figure; subplot(2,1,1); plot(T,Y(:,1),'r','LineWidth',2); set(gca,'FontSize',12); title('Model with mRNA and protein explicitly separated: mRNA'); xlabel('t'); ylabel('Ym'); subplot(2,1,2); plot(T,Y(:,2),'LineWidth',2); set(gca,'FontSize',12); title('Model with mRNA and protein explicitly separated: Protein'); xlabel('t'); ylabel('Y'); functiondy = GeneDE(t,y,p) betam = p(1); alpham = p(2); q = p(3); alpha = p(4); dy = zeros(2,1); dy(1) = betam - alpham*y(1); dy(2) = q*y(1) - alpha*y(2);

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