1. Numerical solution of an ODE system
arising in photosynthesis
Rabindra Gurung
September 10, 2014

2. OUTLINE
Introduction
Stages of photosynthesis
Objectives of dissertation

3. OUTLINE
Introduction
Stages of photosynthesis
Objectives of dissertation
System of differential equations

4. OUTLINE
Introduction
Stages of photosynthesis
Objectives of dissertation
System of differential equations
Steady-states, stability and stiffness of 3 ODEs

5. OUTLINE
Introduction
Stages of photosynthesis
Objectives of dissertation
System of differential equations
Steady-states, stability and stiffness of 3 ODEs
Steady-states, stability and stiffness of 2 ODEs

6. OUTLINE
Introduction
Stages of photosynthesis
Objectives of dissertation
System of differential equations
Steady-states, stability and stiffness of 3 ODEs
Steady-states, stability and stiffness of 2 ODEs
Results between 3 and 2 ODEs

7. OUTLINE
Introduction
Stages of photosynthesis
Objectives of dissertation
System of differential equations
Steady-states, stability and stiffness of 3 ODEs
Steady-states, stability and stiffness of 2 ODEs
Results between 3 and 2 ODEs
Asymptotic analysis of 3 ODEs

8. OUTLINE
Introduction
Stages of photosynthesis
Objectives of dissertation
System of differential equations
Steady-states, stability and stiffness of 3 ODEs
Steady-states, stability and stiffness of 2 ODEs
Results between 3 and 2 ODEs
Asymptotic analysis of 3 ODEs
Sensitivity to the unknown parameters for 3 ODEs

9. OUTLINE
Introduction
Stages of photosynthesis
Objectives of dissertation
System of differential equations
Steady-states, stability and stiffness of 3 ODEs
Steady-states, stability and stiffness of 2 ODEs
Results between 3 and 2 ODEs
Asymptotic analysis of 3 ODEs
Sensitivity to the unknown parameters for 3 ODEs
Comparison of 3 odes solved by Explicit Euler and ode23s

10. OUTLINE
Introduction
Stages of photosynthesis
Objectives of dissertation
System of differential equations
Steady-states, stability and stiffness of 3 ODEs
Steady-states, stability and stiffness of 2 ODEs
Results between 3 and 2 ODEs
Asymptotic analysis of 3 ODEs
Sensitivity to the unknown parameters for 3 ODEs
Comparison of 3 odes solved by Explicit Euler and ode23s
Conclusion and future work

11. Introduction
• Photosynthesis is the process that converts light energy
into chemical energy.
• Photosynthesis removes CO2 and releases O2 into the
atmosphere.
This dissertation is concerned with the numerical solution of a
system of nonlinear ordinary differential equations (ODEs)
which arise in photosynthetic dynamics due to light variation,
which has been investigated experimentally in dark and light
reactions.

12. Stages of photosynthesis
Photosynthesis includes “Light reactions” and “Dark
reactions”.

13. Stages of photosynthesis
Photosynthesis includes “Light reactions” and “Dark
reactions”.
1. Light reactions
• Capture light energy from sunlight.
• Split H2O and produce ATP and NADPH.
• Release O2 as waste product.

14. Stages of photosynthesis
Photosynthesis includes “Light reactions” and “Dark
reactions”.
1. Light reactions
• Capture light energy from sunlight.
• Split H2O and produce ATP and NADPH.
• Release O2 as waste product.
2. Dark reactions
• Does not require light.
• Use ATP and NADPH from CO2.
• Synthesize organic molecules like carbohydrates, glucose.

15. Objectives of dissertation
• to study the steady-states, stability and stiffness of the
system,
• to get numerical solutions using ode solver ode23s,
• to study the initial asymptotic behaviour of the system,
• to compare results solved by ode23s and Explicit Euler
scheme,
• to compare the results of 3 and 2 ODEs model,
• to discuss the sensitivity of unknown parameters of the
system.

16. System of differential equations
• The downstream processes are modelled as
dChlaON
dt
= αIChlaOFF
+ ChlaON
(−kf − kd − knE − kpQ)
• Non photochemical processes are modelled as
dSON
dt
= λbChlaON
SOFF
− λr QSON
• Photochemical processes are modelled as
dQON
dt
= kpQChlaON
− γQON
.

17. Parameters used in system
Table 1: Parameters and their meanings.

18. Non dimensionlization of 3 ODEs
The non-dimensionalized equations of the system of 3 ODEs
dc
dt
= αI(1 − c) + c(−kf − kd − kns − kp(1 − q))
ds
dt
= λbPC c(1 − s) − λr (1 − q) s
dq
dt
= kp(1 − q)
PC
PQ
c − γ q
The non-dimensionalized nonlinear equations were used to
seek the results of the systems.

19. Steady-States of 3 ODEs
Before finding the Steady-States of 3 ODEs, we apply the
Descartes rule of signs into the system of 3 ODEs
• to show the steady-states are positive which lies between
0 and 1.
Then we used Newton Raphson method to find the
steady-states of the system
Table 2: Steady-states of 3 ODEs for c, s and q

20. Stability of 3 ODEs
• We have already determined the steady state values of c,
s, q which we call c, s and q
c = c + δ, s = s + σ, q = q + β
where δ, σ and β are small quantities termed
perturbations.

21. Stability of 3 ODEs
• We have already determined the steady state values of c,
s, q which we call c, s and q
c = c + δ, s = s + σ, q = q + β
where δ, σ and β are small quantities termed
perturbations.
• The eigenvalues of perturbations terms gives negatives.

22. Stability of 3 ODEs
• We have already determined the steady state values of c,
s, q which we call c, s and q
c = c + δ, s = s + σ, q = q + β
where δ, σ and β are small quantities termed
perturbations.
• The eigenvalues of perturbations terms gives negatives.
• Hence stability of the steady-state is stable.

23. Stiffness of 3 ODEs
• We need to find the condition number of Jacobian at the
initial time to know the initial stiffness of the system.
cond(J) ≈ 1012
where cond(J) is max|λJ |
min|λJ |
• The system of 3 ODEs is extremely stiff
Therefore stiff solver ode23s was used to solve the system of
3 ODEs.

24. System of 2 ODEs
Systems of 3 ODEs can be reduced to a systems of 2 ODEs
dc
dt
= αI(1 − c) + c(−kf − kd − kns − kp(1 − q)) (1)
Equation (1) presents a very fast process and is therefore can
be set to zero after a short time. Rearranging the equation
with dc
dt
= 0 for c gives
c =
α I
(αI + kf + kd + kns + kp(1 − q))
.

25. System of 2 ODEs
Substituting the value of c into two remaining equations of
the 3 ODEs system gives
ds
dt
=
αIλbPC (1 − s)
(αI + kf + kd + kns + kp(1 − q))
− λr (1 − q) s
dq
dt
=
kp(1 − q)PC αI
PQ(αI + kf + kd + kns + kp(1 − q))
− γ q.

26. Steady-States and stability of 2 ODEs
• Descartes rule of signs tell us there is a unique steady
states of c, s and q which is positive and lies between 0
and 1

27. Steady-States and stability of 2 ODEs
• Descartes rule of signs tell us there is a unique steady
states of c, s and q which is positive and lies between 0
and 1
• Newton Raphson method to find the steady-states of
2 ODEs

28. Steady-States and stability of 2 ODEs
• Descartes rule of signs tell us there is a unique steady
states of c, s and q which is positive and lies between 0
and 1
• Newton Raphson method to find the steady-states of
2 ODEs
• Just like 3 ODEs the stability of steady states of 2 ODES
was stable.

29. Stiffness of 2 ODEs
• Condition number of Jacobian at the initial time of 2
ODEs
cond(J) ≈ 54
• The system of 2 ODEs is less stiff compared to 3 ODEs
We stiff used stiff solver ode23s to solve the system of 2
ODEs.

30. Results between 3 and 2 ODEs
• Steady-states and stability of 3 and 2 ODEs are same
• Stiffness of 3 and 2 ODEs are different
• Using stiff solver ode23s both systems have same
behaviour of s and q
• The initial behaviour of c are different for both systems
which was solved by ode23s

31. Results between 3 and 2 ODEs
Figure 1: Overplots of 3 and 2 ODEs for c using a final time =0.2
but different tolerance 10−11 (3 ODEs) and 10−4 (2 ODEs).

32. Asymptotic analysis of 3 ODE
• expansion in terms of a small parameters
• first terms of expansion was zero terms
• second terms of expansion was dominant terms
• expansion up to third terms and neglect after that terms

33. Asymptotic analysis of 3 ODE
Figure 2: 3 Asymptotic Expansion and 3 normal ODEs of c, s and
q which shows same.

34. Asymptotic analysis of 3 ODE
Figure 3: 3 Separation point for asymptotic expansion and 3
normal ODEs of c, s and q.

35. Sensitivity to the unknown parameters for 3 ODEs
Three unknown parameters are
1. γ
2. λr
3. λb
• changing values of 3 unknown parameters one at a time
• plot the results of the perturbed solutions with actual
solutions together
• smaller errors in total between the actual and perturbed
values of the parameters
• γ = 2.74, λr = 835 and λb = 0.0087 seems reasonable

36. Comparison of 3 odes solved by Explicit Euler and
ode23s
Figure 4: 3 ODEs solved by Euler Explicit scheme.

37. Comparison of 3 odes solved by Explicit Euler and
ode23s
Figure 5: 3 ODEs solved by ode23s.

38. Conclusion and future work
1. Conclusion
• stiff ODESs was successfully solved numerically
• manage to get same results solved from steady-state
approach and dynamic approach
• asymptotic expansion gave detailed behaviour of the
system close to the initial state

39. Conclusion and future work
1. Conclusion
• stiff ODESs was successfully solved numerically
• manage to get same results solved from steady-state
approach and dynamic approach
• asymptotic expansion gave detailed behaviour of the
system close to the initial state
2. Future work
• asymptotic analysis of 2 ODEs
• to explore more of 2 ODEs in terms of stiffness
• to look at matched asymptotic expansions of 3 ODES