Advances in Numerical Solution of Kinetics Reactions

1
Advances in Numerical Solution of Kinetics
Reactions

• MAMS - Technical Report
• Boanerges Aleman-Meza

2
Agenda

• Introduction
• Numerical Solution
• Graphics of the Solution
• Conclusions

3
Introduction

• Biochemical signaling networks

• Model Representation

Simulation
Model
Refinement
Results Interpretation
Kinetics Solver
4
Basic Model
O
H
H2
O2
2
1
OH
H2 O ? H OH
H O2 ? O OH
3
H2 OH ? H H2O
H2O
5
TransportModel A
6
Reactions ? Equations

• Rate constants
• forward
• backward

kf A B ? C D kb
d( H2 )/dt
-kf1 H2 O
kb1 H OH
-kf3 H2 OH
kb3 H H2O
H2 O ? H OH
H O2 ? O OH
H2 OH ? H H2O
7
Reactions ? Equations

• Rate constants
• forward
• backward

d( O )/dt
-kf1 H2 O
kb1 H OH
kf2 H O2
H2 O ? H OH
-kb2 O OH
H O2 ? O OH
H2 OH ? H H2O
8
Kinetics Solver

• Simulate kinetics reactions like
• H2 O ? H OH
• H O2 ? O OH
• H2 OH ? H H2O

• Initial Time ? Final Time
• Initial Concentrations and rate constants
• Numerical Method selection

9
Numerical Solution
10
Numerical Methods

• Euler Method
• yn1 yn h f(tn, yn)

• Modified Euler
• k0 h f(tn, yn)
• k1 h f(tn, yn k0)
• yn1 yn ½ (k0 k1)

11
Numerical Methods ...

• Fourth order Runge-Kutta
• k1 h f(tn, yn)
• k2 h f(tn ½ h, yn ½ k1)
• k3 h f(tn ½ h, yn ½ k2)
• k4 h f(tn h, yn k3)
• yn1 yn (1/6) (k1 2 k2 2 k3 k4)

12
Numerical Methods ...

• Adaptive Runge-Kutta Fehlberg
• k1 h f(tn, yn)
• k2 h f(tn ¼ h, yn ¼ k1)
• k3 h f(tn c1 h, yn c2 k1 c3 k2)
• k4 h f(tn c4 h, yn c5 k1 c6 k2 c7 k3)
• k5 h f(tn h, yn c8 k1 c9 k2 c10 k3
  c11 k4)
• k6 h f(tn ½ h, yn c12 k1 2 k2 c13 k3
  c12 k4 c13 k5)
• yn1 yn c14 k1 c15 k2 c16 k3 c17
  k4
• yn1 yn c18 k1 c19 k3 c20 k5 c21
  k6

13
Numerical Methods ...

• Backward Difference Formula (Implicit)
• yn1 (4/3) yn - (1/3) yn-1 (2/3) yn1
• Jacobian evaluation
• System of Equations O(m3) O(m2)

14
Performance

• With data set QA-Tr.B
• 39 Species, 45 Reactions

15
Performance

• With data set QA-A
• 37 Species, 43 Reactions

16
Performance

• With data set QA-Tr.-A
• 37 Species, 43 Reactions

17
Performance

• With data set LAC-PTS
• 64 Species, 69 Reactions

18
Graphics of the Solution
19
Using the Kinetics Solver
Model
Kinetics Solver
Input file
Output file
Refinement
Plotting Tool
Plots Web Page
Excel
20
Using the Kinetics Solver ...
Model
Input file
Kin Files Web Page
Refinement
Plot / Save
Kinetics Solver
Plot of the Solution Web Page
Output file
Plot
21
Conclusions
22
Conclusions

• Numerical methods
• performance vs. accuracy
• data set parameters
• - Stiff solver inadequate

• Plotting by Web Page
• Computing Life

23
Questions and Comments!
24
Fortran 77 Code

• Euler method
• Euler modified method

25
Numerical Methods ...

• Parallel Iterated Runge-Kutta
• Fn(0) f(yn)
• Gn(j) A y(n) h B Fn(j-1), j 1,, L
• yn1 0,,0,1T Gn(L)