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Rational Design of Organelle Compartments in Cells Rational Design of Organelle Compartments in Cells Claudio Angione Nettab 2012 Claudio Angione Nettab 2012 | Angione, Carapezza, Costanza, Li o, Nicosia Computer Laboratory, University of

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Rational Design of Organelle Compartments in Cells

Rational Design

  • f Organelle Compartments in Cells

Claudio Angione Nettab 2012

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 1

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Rational Design of Organelle Compartments in Cells

Motivation

Metabolic engineering requires mathematical models for accurate design purposes Aim: overproducing desired substances Problem: identify the interventions needed to produce the metabolite of interest Tools: optimisation, sensitivity, robustness, identifiability

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 2

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Rational Design of Organelle Compartments in Cells

Obstacles

Large number of reactions occurring in the cellular metabolism Large size of the solution space

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 3

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Rational Design of Organelle Compartments in Cells

Idea

We use a multi-objective optimisation algorithm to seek the manipulation that optimise multiple cellular functions The idea is to use and improve the Pareto optimal solutions Pareto optimality is important to obtain not only a wide range of Pareto

  • ptimal solutions, but also the best trade-off design

5 10 15 20 25 30 35 40 45 3.2 3.4 3.6 3.8 4 4.2 4.4

Succinate [mmolh-1 gDW-1] Biomass [h-1]

Anaerobic condition Aerobic condition Pareto Fronts Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 4

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Rational Design of Organelle Compartments in Cells

Outline

Organelle models: Chloroplast model, 31 ODEs + equations for conserved quantities [Zhu et al., 2007] Mitochondrion model, 73 DAEs [Bazil et al., 2010] Hydrogenosome model, Flux Balance Analysis [Angione et al., submitted]

Common framework

1 Sensitivity analysis 2 Multi-objective optimisation 3 Robustness analysis 4 Identifiability analysis

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 5

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Rational Design of Organelle Compartments in Cells

Outline

Organelle models: Chloroplast model, 31 ODEs + equations for conserved quantities [Zhu et al., 2007] Mitochondrion model, 73 DAEs [Bazil et al., 2010] Hydrogenosome model, Flux Balance Analysis [Angione et al., submitted]

Common framework

1 Sensitivity analysis 2 Multi-objective optimisation 3 Robustness analysis 4 Identifiability analysis

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 5

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Rational Design of Organelle Compartments in Cells

Why a common framework for organelles?

All extant eukaryotes are descended from an ancestor that had a mitochondrion The evolutionary history of chloroplasts and mitochondria are intertwined The possibility of multi-objective optimisation related to the different tasks (e.g. maximising the ATP and the heat) Identify and cross-compare the most important components Assess the fragileness of the multi-optimised metabolic networks using the robustness analysis

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 6

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Rational Design of Organelle Compartments in Cells

The common framework

  • metab. 3
  • metab. 4

Sensitivity analysis Multi-objective

  • ptimisation

Robustness analysis

  • metab. 1
  • metab. 2

steady state 1 robust neighborhood (b) trajectories (c) steady state 2 (a) Pareto-optimal

  • rganelle

2 1 3 4 Pareto-optimal

  • rganelles

Model order reduction

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 7

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Rational Design of Organelle Compartments in Cells

Multi-objective optimisation

Let f be the vector of r objective functions to optimise in the objective space

f1 f2 f(y*)

Solution of a multi-objective problem: set of points called Pareto front Represents the best trade-off between two or more requirements A point y ∗ in the solution space is Pareto optimal if there does not exist a point y such that f (y) dominates f (y ∗), i.e. fi(y) > fi(y ∗), ∀i = 1, ..., r

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 8

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Rational Design of Organelle Compartments in Cells

Genetic design through multi-objective optimisation (GDMO) [Costanza et

  • al. Bioinformatics, 2012]

Seek an optimal initial array of concentrations through an evolutionary algorithm inspired by NSGA-II [Deb et al., 2002]

1 generate initial population P(t) 2 evaluate the fitness of each individual in P(t) 3 while (not termination condition) do 1 select parents, Pa(t) from P(t) based on their fitness in P(t) 2 apply crossover to create offspring from parents: Pa(t) -> O(t) 3 apply mutation to the offspring: O(t) -> O’(t) 4 evaluate the fitness of each individual in O’(t) 5 select population P(t+1) from current offspring O’(t) and parents Pa(t)

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 9

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SLIDE 11

Rational Design of Organelle Compartments in Cells

Genetic design through multi-objective optimisation (GDMO) [Costanza et

  • al. Bioinformatics, 2012]

Seek an optimal initial array of concentrations through an evolutionary algorithm inspired by NSGA-II [Deb et al., 2002]

1 generate initial population P(t) 2 evaluate the fitness of each individual in P(t) 3 while (not termination condition) do 1 select parents, Pa(t) from P(t) based on their fitness in P(t) 2 apply crossover to create offspring from parents: Pa(t) -> O(t) 3 apply mutation to the offspring: O(t) -> O’(t) 4 evaluate the fitness of each individual in O’(t) 5 select population P(t+1) from current offspring O’(t) and parents Pa(t)

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 9

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SLIDE 12

Rational Design of Organelle Compartments in Cells

Genetic design through multi-objective optimisation (GDMO) [Costanza et

  • al. Bioinformatics, 2012]

Seek an optimal initial array of concentrations through an evolutionary algorithm inspired by NSGA-II [Deb et al., 2002]

1 generate initial population P(t) 2 evaluate the fitness of each individual in P(t) 3 while (not termination condition) do 1 select parents, Pa(t) from P(t) based on their fitness in P(t) 2 apply crossover to create offspring from parents: Pa(t) -> O(t) 3 apply mutation to the offspring: O(t) -> O’(t) 4 evaluate the fitness of each individual in O’(t) 5 select population P(t+1) from current offspring O’(t) and parents Pa(t)

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 9

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Rational Design of Organelle Compartments in Cells

Model Reduction and Sensitivity Analysis

State space

steady state

Meta-model construction ARD Sensitivity

initial condition trajectory

State meta-space

steady state initial condition trajectory

Parameter space

parameter values

Parameter space

neighbourhood of parameter values

Organelle complete model State space reflects metabolism Very accurate High-dimensional parameter space Computationally expensive to analyse Organelle metamodel Approximation of the real model Easy to analyse Investigate the sensitivity and robustness Same initial condition Slightly different trajectories

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 10

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Rational Design of Organelle Compartments in Cells

Multi-objective Optimisation and Robustness Analysis

metabolite 1 metabolite 2 multi-objective

  • ptimization

State space

steady state 1 robust neighbourhood (b) trajectories (c)

Robustness Analysis

metabolite 1 metabolite 2 Pareto-optimal

  • rganelle

steady state 2 (a)

  • ptimal

Pareto-front

Multi-objective optimisation Move the front towards the best Pareto-front Maximise metabolites (e.g. ATP vs NADH) Choose Pareto-optimal organelle Robustness of the Pareto optimal organelle (a) Maintains its functionality if it transits through a new steady state [Kitano, 2007] (b) Robustness to change of initial conditions [Gunawardena, 2009] (c) Percentage of perturbation trials such that the output remains in a given interval [Stracquadanio & Nicosia, 2011]

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 11

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Rational Design of Organelle Compartments in Cells

ARD Sensitivity and Reduction in the Mitochondrion Metamodel

0.5 1 1.5 2 x 10

8

5 10 15 x 10

5

−15 −10 −5 5 F1FO Plot of atp_output using PolynomialModel (built with 1028 samples) HK atp_output

Metamodel Closer look at the model behaviour Polynomial surrogate models 1028 samples in the parameter space Second order model: a0 + c⊺p + p⊺Ap p = array of parameters Most sensitive parameters: Hexokinase max rate (HK), F1F0 ATP synthase activity Low values of HK: changes in F1F0 have little effect on the ATP production High values of HK: the mitochondrion is highly sensitive to variations of F1F0

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 12

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Rational Design of Organelle Compartments in Cells

Multi-objective Optimisation in the Chloroplast Model

50000 100000 150000 200000 250000 300000 5 10 15 20 25 30 35 40 45

Total Nitrogen Concentration (mg l-1)

CO2 Uptake (µmol m-2 s-1) Feasible not pareto optimal Chloroplasts Natural Chloroplast Pareto optimal Chloroplasts

CO2 uptake rate vs. protein nitrogen consumption Sensitive domain: 11 most sensitive enzymes Multi-objective optimisation in the “sensitive domain” ( The other 12 enzymes kept at their nominal value Goal: Higher CO2 uptake employing less nitrogen Absorbing more CO2 while consuming less “leaf-fuel” Find all those sensitive enzyme concentration vectors ˆ x = (c1, c2, . . . , c11) such that the resulting CO2 uptake function is maximised and the nitrogen consumption is minimised. This renders the metabolism cycle more efficient. max

ˆ x∈R11 (f1(ˆ

x), −f2(ˆ x))T f1 = CO2 uptake, f2 = nitrogen consumption

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 13

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Rational Design of Organelle Compartments in Cells

Multi-objective Optimisation in the Mitochondrion Model

  • 2e-006

2e-006 4e-006 6e-006 8e-006 1e-005

  • 4000
  • 3000
  • 2000
  • 1000

1000 2000 3000

NADH [nmol/mg] ATP [nmol/mg]

Ca2+ = 10e-6 nmol/mg Ca2+ = 10e-5 nmol/mg Ca2+ = 10e-6/1.5 nmol/mg 0.002 0.004 0.006 0.008 0.01 0.012 0.014

  • 4000 -3000 -2000 -1000

1000 2000 Ca2+ = 10e-6*1.5 nmol/mg Ca2+ = 10e-7 nmol/mg

NADH vs. ATP Pareto fronts Different Ca2+ concentrations Goal: Higher ATP and NADH Genetic algorithm to move the Pareto-front Before the optimisation NADH = 1.5987 · 10−10 nmol/mg (formation) ATP = −0.0014 nmol/mg (consumption) f1 = ATP production, f2 = NADH production

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 14

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Rational Design of Organelle Compartments in Cells

Multi-objective Optimisation in the Hydrogenosome Model

100 200 300 100 200 300

CO2_out

20 40 60 80

  • H2_out

100 300 500

  • ATP_out

100 200 300 50 150 250

  • 20

40 60 80

  • 100

300 500

  • 50

150 250 50 150 250

NADH_out

Hydrogenosome Pareto front MCMC sampling of the reaction network Trade-offs among the maximisations Most reactions are coupled NADH and H2 are in contrast H2 and CO2 seem uncorrelated Red points are the optimal points CO2-NADH plot: the hydrogenosome is versatile and can produce both, but it cannot specialise in producing only one metabolite (higher curvature of the front).

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 15

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Rational Design of Organelle Compartments in Cells

Robustness Analysis

Assess the ability of a system to preserve its behaviour despite internal or external perturbations Perturbation γ (Ψ, σ): applies a stochastic noise σ to the system Ψ Generate a set T of trial samples τ = γ (Ψ, σ) An element τ ∈ T is said to be robust to the perturbation [Stracquadanio & Nicosia, 2011], due to stochastic noise σ, for a given property (or metric) φ, if: ρ (Ψ, τ, φ, ǫ) =

  • 1

if |φ (Ψ) − φ (τ) | ≤ ǫ

  • therwise,

where Ψ is the reference system, ǫ is a robustness threshold. Robustness of a system Ψ: the percentage of robust trials Γ (Ψ, T, φ, ǫ) =

  • τ∈T ρ (Ψ, τ, φ, ǫ)

|T| .

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 16

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Rational Design of Organelle Compartments in Cells

Robustness Analysis

Assess the ability of a system to preserve its behaviour despite internal or external perturbations Perturbation γ (Ψ, σ): applies a stochastic noise σ to the system Ψ Generate a set T of trial samples τ = γ (Ψ, σ) An element τ ∈ T is said to be robust to the perturbation [Stracquadanio & Nicosia, 2011], due to stochastic noise σ, for a given property (or metric) φ, if: ρ (Ψ, τ, φ, ǫ) =

  • 1

if |φ (Ψ) − φ (τ) | ≤ ǫ

  • therwise,

where Ψ is the reference system, ǫ is a robustness threshold. Robustness of a system Ψ: the percentage of robust trials Γ (Ψ, T, φ, ǫ) =

  • τ∈T ρ (Ψ, τ, φ, ǫ)

|T| .

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 16

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SLIDE 21

Rational Design of Organelle Compartments in Cells

Robustness Analysis

Assess the ability of a system to preserve its behaviour despite internal or external perturbations Perturbation γ (Ψ, σ): applies a stochastic noise σ to the system Ψ Generate a set T of trial samples τ = γ (Ψ, σ) An element τ ∈ T is said to be robust to the perturbation [Stracquadanio & Nicosia, 2011], due to stochastic noise σ, for a given property (or metric) φ, if: ρ (Ψ, τ, φ, ǫ) =

  • 1

if |φ (Ψ) − φ (τ) | ≤ ǫ

  • therwise,

where Ψ is the reference system, ǫ is a robustness threshold. Robustness of a system Ψ: the percentage of robust trials Γ (Ψ, T, φ, ǫ) =

  • τ∈T ρ (Ψ, τ, φ, ǫ)

|T| .

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 16

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Rational Design of Organelle Compartments in Cells

Robustness of Enzymes in Natural Chloroplast

Figure: The chloroplast is robust to perturbations of the enzyme concentration if the CO2 uptake rate

is close to the nominal value (15.48µmol/m2s) in the majority of the perturbation trials.

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 17

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Rational Design of Organelle Compartments in Cells

Identifiability Analysis in Chloroplast

1 2 3 4 5 6 2 4 6 0.5 1 1.5 GAPDH Rubisco FBPase

1 2 3 4 5 6 −2 2 Rubisco β 1 2 3 4 5 −2 2 GAPDH β 0.5 1 1.5 −0.2 0.2 FBPase β

Detect relations among decision variables of the optimisation Structural non-identifiability: functional relation among decision variables

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 18

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Rational Design of Organelle Compartments in Cells

Network of Organelles

Systems composed of different species living and interacting in the same

  • rganism

Design, analyse and optimise the “global” metabolism Optimise two or more objectives in different organelles simultaneously Highlight the complementarity of different metabolisms

Example

Mitochondria and chloroplasts are (usually) both found in plants Part of the same functional pipeline Starting from CO2, the photosynthesis in the chloroplast creates glucose that enters the mitochondrion to create ATP

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 19

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SLIDE 25

Rational Design of Organelle Compartments in Cells

Network of Organelles

Systems composed of different species living and interacting in the same

  • rganism

Design, analyse and optimise the “global” metabolism Optimise two or more objectives in different organelles simultaneously Highlight the complementarity of different metabolisms

Example

Mitochondria and chloroplasts are (usually) both found in plants Part of the same functional pipeline Starting from CO2, the photosynthesis in the chloroplast creates glucose that enters the mitochondrion to create ATP

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 19

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Rational Design of Organelle Compartments in Cells

Evolution through Pareto fronts

metabolite 1 metabolite 2 metabolite 1 metabolite 2 Ancestor (bacterium) Organelle 1 Organelle 2 Eukaryotic cell metabolite 1 metabolite 2 Aggregate Pareto front Organelle 1 – Organelle 2 metabolite 1 metabolite 2

The evolution of a Pareto-front can highlight the benefits of an engulfment and subsequent compartmentalisation Pareto-optimal point before the engulfments outperformed by the aggregate Pareto-optimal point.

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 20

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Rational Design of Organelle Compartments in Cells

Conclusion

1 Pareto fronts combined with sensitivity, robustness and identifiability 2 Understand the steps of the cellular evolution and the engulfments and

specialisation of organelles

3 In silico design to explore the reaction network for the solutions that optimise

two or more objectives simultaneously

4 Comprehensive insight into the energy balance in the cell 5 Possible explanation of evolution and compartmentalisation

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 21

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Rational Design of Organelle Compartments in Cells

Acknowledgements

Giovanni Carapezza, Dept. of Maths and Computer Science, University of Catania Jole Costanza, Dept. of Maths and Computer Science, University of Catania

  • Dr. Pietro Li´
  • , Computer Laboratory, University of Cambridge
  • Dr. Giuseppe Nicosia, Dept. of Maths and Computer Science, University of

Catania

Claudio Angione | Angione, Carapezza, Costanza, Li´

  • , Nicosia

Computer Laboratory, University of Cambridge (UK) Nettab 2012 | 22