VERIFICATION OF ROBUSTNESS PROPERTY IN CHEMICAL REACTION NETWORKS

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VERIFICATION OF ROBUSTNESS PROPERTY IN CHEMICAL REACTION NETWORKS

Lucia Nasti

Ph.D. Student

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OUTLINE

➤ What is robustness?

➤ Formalisation problem: CRN and Petri Nets

➤ Why and how to study monotonicity in CRN?

➤ Results: Sufficient conditions and Tool

➤ Application: Becker-Döring equations

➤ Future work

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BACKGROUND

➤ A cell is a very complex system

➤ Chemical reaction networks

(pathways) govern the basic cell’s activities

➤ To examine the structure of the cell as a whole, we can design

multiscale and predictive models

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ROBUSTNESS PROPERTY

➤ Robustness: A fundamental feature of complex evolving systems, for which the behaviour of the system remain essentially constant, despite the presence of internal and external perturbations.

In nature, there are different mechanisms ensuring this property:

➤ Modularity: many components

➤ System control: amplification of the input signal

➤ Redundancy: many structures for the same function

➤ Structural stability: adaptation to external input

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S

E R ₁

ES R 3 P

R 21 R 23

+ +

R ₂

OUR NEW DEFINITION OF INITIAL CONCENTRATION ROBUSTNESS

Input

Output

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➤ Using Petri net → Formal definition of 𝛼 -Robustness and β-Robustness

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S

E R ₁

ES R 3 P

R 21 R 23

+ +

R ₂

OUR NEW DEFINITION OF INITIAL CONCENTRATION ROBUSTNESS

➤ Using Petri net → Formal definition of 𝛼 -Robustness and β-Robustness

[min, max]

[min, max] ss

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EXAMPLE OF APPLICATION OF OUR DEFINITION: CHEMOTAXIS OF E. COLI

➤ Given a set of reactions: ^➤ We build the Petri net:

Input Input

Output

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CHEMOTAXIS OF E.COLI: SIMULATION RESULTS

We vary the initial concentration of the inputs ([R]) and we obtain these concentrations for the species [Yp]. Hence, we obtain 𝛼=0.5 and β=0.35.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Concentration

Yp

Initial concentrations:

L=100 R=1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Concentration

Yp

Initial concentrations:

L=100 R=100

PROBLEM:

TOO MANY SIMULA

TIONS!

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HOW TO LIMIT THE COMPUTATIONAL EFFORT OF SIMULATIONS?

MONOTONICITY

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MONOTONICITY IN CRN

➤ In [Angeli et al., 2008]:

1. Very strong notion of monotonicity: each species have to increase or decrease continually

2. This notion of monotonicity work on particular chemical reaction networks 3. To provide graphical conditions to check global monotonicity:

The system is orthant-monotone if the associated R-graph is sign consistent, hence when any loop has an even number of negative edges.

S

E

R 1 ES R 2 P

S

E

R 1 ES R 2 P

R 1 R 2

SR-GRAPH R-GRAPH

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INPUT-OUTPUT MONOTONICITY

➤ Positive Input-Output Monotonicity. Given a set of reactions R, species O is positively monotonic w.r.t I ∈ R iff, ∀ Ī≥I, Ō≥O, for every time t ∈ ℝ _≥0 .

➤ Negative Input-Output Monotonicity. Given a set of reactions R, species O is negatively monotonic w.r.t I ∈ R iff, ∀ Ī≥I, Ō≤O, for every time t ∈ ℝ _≥0 .

➤ A consistent labelling of a signed graph (V _R , E ₊ , E _- ) is a labelling s: V →{+,-} in which vertices R _i , R _j ∈ V _R have the same label if R _i , R _j ∈ E ₊ , and opposite labels if R _i , R _j ∈ E _-

S

E

R ₁ ES R ₂ P

R R

S

E

R 1 ES R 2 P

+ +

LR-GRAPH

R-GRAPH

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OUR RESULT: INPUT-OUTPUT MONOTONICITY THEOREM

➤ Theorem. Let a set of chemical reactions G be given, with I and O as input and output species. If the following three conditions hold:

1. the R-graph of G has the positive loop property and hence admits a consistent labelling s;

2. The species I participates in only one reaction R _I ;

3. The species O participates in only one reaction R _O.

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INPUT-OUTPUT MONOTONICITY: MICHAELIS MENTEN KINETICS

STOICHIOMETRIC MATRIX

S

E

R 1 ES R 2 P

R ₁ R ₂

+ +

LR-GRAPH

0 200 400 600 800 1000 1200 1400 1600 1800 2000

[S] 0

0 5 10 15 20 25 30 35 40 45 50

[P] ss

P

SIMULATION RESULT

USING OUR

THEOREM WE NEED JUST 1 SIMULA

TION!

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➤ Tool (in Python) to verify our sufficient conditions on big graphs

INPUT-OUTPUT GRAPHTOOL

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➤ It is a model that describes condensations phenomena at different pressures

➤ The clusters give rise to two types of reactions:

BECKER-DÖRING MODEL

where:

➤ C i denotes clusters consisting of i particles

➤ Coefficients a i and b i+1 stand, respectively, for the rate of aggregation and fragmentation

➤ Rates may depend on the size of clusters involved in the reactions

➤ The mass is constant and it depends on the initial condition of the system

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STEADY STATE ANALYSIS

➤ Theorem 3. Let a and b be the coefficient rates of coagulation and

fragmentation process in the Becker-Döring system, ρ the mass of the system and [C 1 ] ss the concentration of monomers at the steady state.

Then, as ρ →∞, [C 1 ] ss → .

➤ With rates a=b, changing the initial concentration of C 1 , the monomer concentration at the steady state tends to 1

0.5 1 1.5

[C 1 ] ss

C ₁

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RESULTS

➤ Formal definition of absolute and relative concentration robustness

➤ Analysis of the systems by simulations

➤ Sufficient conditions to study monotonicity between Input and Output species

➤ Implementation of Input-Output GraphTool

➤ Verification of Robustness of Becker-Döring equations

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PROPOSED RESEARCH PROJECT

1. Stochasticity

2. Investigation of other topological features 3. Applicability to new specific problems

VERIFICATION OF ROBUSTNESS PROPERTY IN CHEMICAL REACTION NETWORKS