1. Dynamic Systems: Ordinary Differential Equations
BME383J

2. Objectives Model the dynamics of bio systems with ODE
Analytical solutions (brief)
Numerical solutions
Accuracy and stability of numerical solutions

3. Pharmacokinetics Administration: Oral Intravenous injection
Absorption:
dissolution rate
Blood flow
Permeation/ionization
Drug affinity
Distribution:
Compartments
Metabolism:
Organ/Cell/Enzyme
Elimination:
Kidney

4. Ion transport through Membrane
Hodgkin-Huxley model of Na+ conduction
g: conductance, V: potential difference, I: current;
n gates open for K+; m and h gates for Na+

5. Classification of ODE Order n (dny/dtn) Linear or nonlinear
Nonlinear: powers of y or the derivatives, or products of y with the derivatives.
Homogeneous or nonhomogeneous
f(y, dny)=R(t)
Second-order, linear, nonhomogeneous

6. What are these? First order, nonlinear, nonhomogeneous.
3rd order, linear, homo
3rd, nonlinear, nonhomo.
2nd order, nonlinear.

7. Boundary conditions initial-value problems boundary-value problems
the values of the y and/or dy are all known at the initial (or final) value of t
boundary-value problems
y and/or dy are known at more than one point of t.
E.g. y(t=0) and dy/dt(t=final) are known: two-point boundary value problem.
“Numerical methods for chemical engineers with MATLAB applications”, Constantinides and Mostoufi (1999)

8. Canonical Form ODE
a set of n simultaneous first-order ordinary differential equations
Matrix Notation
When the initial conditions are given at a common point, to
Solution of above eq:

9. Converting high order to 1st order ODE
First derivative of z
First derivative of y1
Second derivative of z
First derivative of y2
First derivative of yn

10. z=y1
y5=e-t

11. Matrix Formula
Y’=AY

12. Analytical Solutions Separable variables Example: y=__________________
Ln y = lambda t + C
Y= C exp (lambda t)
Example:
y=__________________

13. Matlab dsolve
YY=dsolve('Dy=A*y')

14. Analytical solution
Equations with homogeneous coefficients and same degree
Replace one variable to separate variables
Dt/t+vdv/(v-1)=0
W=lambert(x) is the solution to W*exp(W)=x

15. Analytical Solution
Exact solution

16. Analytical Solution
Exact solution

17. Analytical Solution
Integrating factor
Homework
Compare with dsolve

18. Analytical Solution Linear homogeneous eq. with constant coeff.
Characteristic equation of D
If the roots, λ are real
If the roots are complex

19. Analytical Solution Show that Compare with desolve
1*D^3-9*D^2+26*D-24=0
p=[ ]
roots{p}
y=dsolve('D3y-9*D2y+26*Dy-24*y=0')

20. Numerical Solution of Nonlinear ODE
y at different t?

21. Euler Methods
Forward marching formula

22. Mixing Forward and Backward Euler Method for Better Accuracy
Backward Taylor series
Implicit or backward Euler method:
unknown
First apply the forward/explicit Euler to predict
Forward or backward is determined by the derivative at I or i+1…
O(h2) has opposite sign so they cancel!
Then use the backward Euler to correct:
Euler predictor-corrector method: O(h3)

23. Numerical Integration of ODE Runge-Kutta

24. Numerical Integration of ODE Runge-Kutta

25. Runge-Kutta
Second order
Fourth order
Third order

26. Alternative: Verlet Integration
Modeling trajectories of particles after Newton’s equation of motion

27. Verlet Integration
Order of magnitude more accurate than straight Taylor expansion

28. Verlet Variations

29. RK4 conserves the dxdp better
VV conserves the energy better

30. MATLAB ODE
A stiff equation is a differential equation for which certain numerical methods for solving the equation are unstable, unless the step size is taken to be extremely small.
ODE45 in matlab is good for nonstiff
ODE23s for stiff
What to use

31. ODE numerical solver in MATLAB

32. Example For a solution for and initial conditions: y1(0)=0, y2(0)=3
yprime.m
function dydt=yprime(t,y) %
dydt=[y(2)
-0.2*y(2)-sin(y(1))];
%Or write the elements explicitly:
%dydt(1,1)=y(2); dydt(2,1)=-0.2*y(2)-sin(y(1));
[t, [0 40], [0 3]);
For a solution for and initial conditions: y1(0)=0, y2(0)=3
[t, y]=ode45(____ , ____ , ____)

33. ODE45 with a free parameter a
The [] in the list of arguments for yprime is an empty placeholder where certain options can be input to ode45.

34. Example of Enzyme Catalysis
Mass Balance
Assuming first order reaction w.r.t. each reach reactant
Assumption?

35. MATLAB
Enzyme_7_2.m
Enzyme_kinetics_equations.m

36. Solution to Linear ODE Method based on Eigenvalues and Eigenvectors
Recall “variable separation”

37. Solution to multiple Linear ODE
In matrix form
Proof:
F(x)=f(0)+(x-0)f’(0)/1!+(x-0)^2f”(x)/2!+(x=0)^ifi(0)/i!

38. Solution to multiple Linear ODE
If A has eigenvalues Λ and eigenvectors X
AX=XΛ
X=[x1, x2, x3, …, xn]
Derivation see Constantinides and Mostoufi (1999) Numerical Methods for Chemical Engineers with MATLAB Applications.

39. Matlab Functions for matrix exponentials and eigenvalues/eigenvectors:
expm (A): Calculates the matrix exponential of A using a scaling and squaring algorithm with a Pade approximation (Burden et ai., 1981).
eig (A) : Calculates the eigenvalues of matrix A.
[X, LAMBDA] = eig (A)
y = X*expm(LAMBDA*t)*X^-1*y0

40. Example: The dynamics of drug absorption

41. Matlab code disp('Initial concentrations: ') c0= [1; 0; 0]
disp(' '); disp('Matrix of coefficients: ')
K=[-k ; k0 -k1 0; 0 k1 0]
[X,lambda]=________;
c=___*expm(_____)*___ *c0
t=[0:100];
c=eval(c);
disp('Initial concentrations: ')
c0= [1; 0; 0]
disp(' '); disp('Matrix of coefficients: ')
K=[-k ; k0 -k1 0; 0 k1 0]
[X,lambda]=eig(K);
c=X*expm(lambda*t)*X^-1*c0
t=[0:100];
c=eval(c);

42. Steady-State Solutions and Stability Analysis
Steady-state: time derivative is zero

43. Steady-State Solutions and Stability Analysis
At steady-state:

44. Introducing perturbation

45. Jacobian: Solution depends on the eigenvalues of A (J*):
Recall linear ODE:
Solution depends on the eigenvalues of A (J*):

46. Stability Analysis
Analogously, in 1-D, gradients inform us about minimum, maximum or saddle point.

50. Negative real part
Positive real part
Nonzero complex part

51. Numerical Stability Stability:
Inherent (positive real in eigenvalues of J matrix)
Numerical (integration method)
Truncation error
Round off error
Propagation error
More terms in the expansion (smaller steps)
Double precision
Smaller steps

52. Stability of Euler Method
Initial value differential equation
Analytical solution
Is inherently stable for
because
Introduce a small perturbation to the initial value:
If λ > 0, small changes in initial value lead to small changes in solution: unstable

53. Numerical Stability of Euler Method
Numerical solution by explicit Euler:
The numerical solution is absolutely stable if
This requires the growth factor
Euler method is conditionally stable within the linear stability domain.

55. Stiff Equations
A stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small.
The MATLAB functions ode23s and ode15s are suitable for stiff ordinary differential equations

56. Stiff Ratio
For a set of nonlinear DE:
Big SR  Stiff

57. Stiff Equation Example: The chemical reaction of Robertson
With a short interval, e.g. (0<h<40) numerical integration is stable.
However, if h is very large (1011), then many standard codes fail.

58. Matlab Exercise
Select one book example below, work through and present in class.
Example 7.4a (or b), 7.5, 7.6, 7.7. Scanned PDF on BlackBoard.

59. Lessons Learned
Molecular, cellular and bio systems can be modeled with ODE
ODE classification:
1st, 2nd, 3rd order
Linear, nonlear
Homogenous, nonhomogeneous
Initial value or boundary value
2nd or higher order ODE can be converted to a set of 1st order ODE
Linear ODE solution depends on eigenvalues and eigenvectors of the ODE (expam)
Nonlinear ODE can be integrated numerically using method based on FD.
Integrating ODE is line climbing a mountain: move in the direction of slope (or weighted average), taking many small steps.
Solve ODE using MATLAB (ode45, expam)
The stability of nonlinear ODE depends on the eigenvalues and the Jacobian matrix
The stability of numerical solution depends on the form of the equations, the methods, and step size.