Numerical solution of an ODE system

arising in photosynthesis

Rabindra Gurung

September 10, 2014

OUTLINE

IntroductionStages of photosynthesisObjectives 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

OUTLINE

IntroductionStages of photosynthesisObjectives 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

OUTLINE

IntroductionStages of photosynthesisObjectives 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

OUTLINE

IntroductionStages of photosynthesisObjectives 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

OUTLINE

IntroductionStages of photosynthesisObjectives 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

OUTLINE

IntroductionStages of photosynthesisObjectives 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

OUTLINE

IntroductionStages of photosynthesisObjectives 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

OUTLINE

IntroductionStages of photosynthesisObjectives 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

OUTLINE

IntroductionStages of photosynthesisObjectives 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

Introduction

• Photosynthesis is the process that converts light energyinto chemical energy.

• Photosynthesis removes CO2 and releases O2 into theatmosphere.

This dissertation is concerned with the numerical solution of asystem of nonlinear ordinary differential equations (ODEs)which arise in photosynthetic dynamics due to light variation,which has been investigated experimentally in dark and lightreactions.

Stages of photosynthesis

Photosynthesis includes “Light reactions” and “Darkreactions”.

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.

Stages of photosynthesis

Photosynthesis includes “Light reactions” and “Darkreactions”.

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.

Stages of photosynthesis

Photosynthesis includes “Light reactions” and “Darkreactions”.

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.

Objectives of dissertation

• to study the steady-states, stability and stiffness of thesystem,

• 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 Eulerscheme,

• to compare the results of 3 and 2 ODEs model,

• to discuss the sensitivity of unknown parameters of thesystem.

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= λbChla

ONSOFF − λrQSON

• Photochemical processes are modelled as

dQON

dt= kpQChla

ON − γQON .

Parameters used in system

Table 1: Parameters and their meanings.

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

PQc − γ q

The non-dimensionalized nonlinear equations were used toseek the results of the systems.

Steady-States of 3 ODEsBefore finding the Steady-States of 3 ODEs, we apply theDescartes rule of signs into the system of 3 ODEs

• to show the steady-states are positive which lies between0 and 1.

Then we used Newton Raphson method to find thesteady-states of the system

Table 2: Steady-states of 3 ODEs for c, s and q

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 termedperturbations.

• The eigenvalues of perturbations terms gives negatives.

• Hence stability of the steady-state is stable.

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 termedperturbations.

• The eigenvalues of perturbations terms gives negatives.

• Hence stability of the steady-state is stable.

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 termedperturbations.

• The eigenvalues of perturbations terms gives negatives.

• Hence stability of the steady-state is stable.

Stiffness of 3 ODEs

• We need to find the condition number of Jacobian at theinitial 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 of3 ODEs.

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 canbe set to zero after a short time. Rearranging the equationwith dc

dt= 0 for c gives

c =α I

(αI + kf + kd + kns + kp(1 − q)).

System of 2 ODEs

Substituting the value of c into two remaining equations ofthe 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)PCαI

PQ(αI + kf + kd + kns + kp(1 − q))− γ q.

Steady-States and stability of 2 ODEs

• Descartes rule of signs tell us there is a unique steadystates of c, s and q which is positive and lies between 0and 1

• Newton Raphson method to find the steady-states of2 ODEs

• Just like 3 ODEs the stability of steady states of 2 ODESwas stable.

Steady-States and stability of 2 ODEs

• Descartes rule of signs tell us there is a unique steadystates of c, s and q which is positive and lies between 0and 1

• Newton Raphson method to find the steady-states of2 ODEs

• Just like 3 ODEs the stability of steady states of 2 ODESwas stable.

Steady-States and stability of 2 ODEs

• Descartes rule of signs tell us there is a unique steadystates of c, s and q which is positive and lies between 0and 1

• Newton Raphson method to find the steady-states of2 ODEs

• Just like 3 ODEs the stability of steady states of 2 ODESwas stable.

Stiffness of 2 ODEs

• Condition number of Jacobian at the initial time of 2ODEs

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 2ODEs.

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 samebehaviour of s and q

• The initial behaviour of c are different for both systemswhich was solved by ode23s

Results between 3 and 2 ODEs

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

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

Asymptotic analysis of 3 ODE

Figure 2: 3 Asymptotic Expansion and 3 normal ODEs of c , s andq which shows same.

Asymptotic analysis of 3 ODE

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

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 actualsolutions together

• smaller errors in total between the actual and perturbedvalues of the parameters

• γ = 2.74, λr = 835 and λb = 0.0087 seems reasonable

Comparison of 3 odes solved by Explicit Euler and

ode23s

Figure 4: 3 ODEs solved by Euler Explicit scheme.

Comparison of 3 odes solved by Explicit Euler and

ode23s

Figure 5: 3 ODEs solved by ode23s.

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

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