structural stability of biochemical networks: …

8th IFAC Symposium on Robust Control Design ROCOND’15Bratislava, Slovenská Republika

July 10, 2015

STRUCTURAL STABILITY OFBIOCHEMICAL NETWORKS:

QUADRATIC VS. POLYHEDRALLYAPUNOV FUNCTIONS

Franco Blanchini and Giulia Giordano

A powerful insight into the features of living matter

Biological systems are extremely robust:fundamental properties are always preserved

despite huge uncertainties and parameter variations

STRUCTURAL ANALYSIS

Can we explain behavioursbased on the systeminherent structure only?

structure graph

motif ←→ structural property

Franco Blanchini, Giulia Giordano Lyapunov functions for structural stability of biosystems

STRUCTURAL ANALYSIS

structure graph

STRUCTURAL ANALYSIS

structure graph

STRUCTURAL ANALYSIS

structure graph

Structural: more than robust

F family of systems, P property

Robust propertyP is robust if any element f ∈ F has the property P.

Structural propertyP is structural if, moreover, F is specified by a “structure”without numerical bounds.

Robust vs. Structural stability

[−a bc −d

], A2 =

[−a −bc −d

], a, b, c , d > 0.

A1 is robustly stable if 0 ≤ b, c ≤ 1, 2 ≤ a, d ≤ 3.A2 is structurally stable.

[−a bc −d

], A2 =

[−a −bc −d

], a, b, c , d > 0.

[−a bc −d

], A2 =

[−a −bc −d

], a, b, c , d > 0.

[−a bc −d

], A2 =

[−a −bc −d

], a, b, c , d > 0.

(Bio)chemical reaction networks I

Species: A, B , C , . . .

Reactions:

∅ a0−⇀ A, ∅ b0−⇀ B, A + Bgab−−⇀↽−−gc

C , Cgc−⇀ D, D

gd−⇀ E , A + Egae−−⇀ ∅

Concentrations: a, b, c , . . .

ODE system:

a = a0−gab(a, b) + gc(c)−gae(a, e)

b = b0−gab(a, b) + gc(c)

c = gab(a, b)− gc(c)−gc(c)

d = gc(c)−gd(d)

e = gd(d)−gae(a, e)

Reactions:

∅ a0−⇀ A, ∅ b0−⇀ B

, A + Bgab−−⇀↽−−gc

C , Cgc−⇀ D, D

gd−⇀ E , A + Egae−−⇀ ∅

ODE system:

a = a0

−gab(a, b) + gc(c)−gae(a, e)

b = b0

−gab(a, b) + gc(c)

gab(a, b)− gc(c)−gc(c)

gc(c)−gd(d)

gd(d)−gae(a, e)

Reactions:

∅ a0−⇀ A, ∅ b0−⇀ B, A + Bgab−−⇀↽−−gc

, Cgc−⇀ D, D

gd−⇀ E , A + Egae−−⇀ ∅

ODE system:

a = a0−gab(a, b) + gc(c)

−gae(a, e)

c = gab(a, b)− gc(c)

−gc(c)

gc(c)−gd(d)

gd(d)−gae(a, e)

Reactions:

∅ a0−⇀ A, ∅ b0−⇀ B, A + Bgab−−⇀↽−−gc

C , Cgc−⇀ D

, Dgd−⇀ E , A + E

gae−−⇀ ∅

ODE system:

−gae(a, e)

d = gc(c)

−gd(d)

gd(d)−gae(a, e)

Reactions:

∅ a0−⇀ A, ∅ b0−⇀ B, A + Bgab−−⇀↽−−gc

C , Cgc−⇀ D, D

gd−⇀ E

, A + Egae−−⇀ ∅

ODE system:

−gae(a, e)

d = gc(c)−gd(d)

e = gd(d)

−gae(a, e)

Reactions:

∅ a0−⇀ A, ∅ b0−⇀ B, A + Bgab−−⇀↽−−gc

C , Cgc−⇀ D, D

gd−⇀ E , A + Egae−−⇀ ∅

ODE system:

a = a0−gab(a, b) + gc(c)−gae(a, e)

d = gc(c)−gd(d)

e = gd(d)−gae(a, e)

(Bio)chemical reaction networks II

ODE system:

−1 −1 1 0 00 −1 1 0 00 1 −1 −1 00 0 0 1 −1−1 0 0 0 1

︸︷︷︸

S = stoichiometric matrix

gae(a, e)gab(a, b)gc(c)gc(c)gd(d)

︸︷︷︸

g = rate

a0b0000

︸︷︷︸

g0 = influx

Graph:

ODE system:

−1 −1 1 0 00 −1 1 0 00 1 −1 −1 00 0 0 1 −1−1 0 0 0 1

︸︷︷︸

g = rate

a0b0000

︸︷︷︸

g0 = influx

Graph:

ODE system:

−1 −1 1 0 00 −1 1 0 00 1 −1 −1 00 0 0 1 −1−1 0 0 0 1

︸︷︷︸

g = rate

a0b0000

︸︷︷︸

g0 = influx

Graph:

General nonlinear model

Nonlinear model

x(t) = Sg(x(t)) + g0

S stoichiometric matrix

Reaction rate functions g : nonnegative and monotonicConstant influx vector g0 ≥ 0Dissipative reactions: ∂xi

∂xi< 0

Positive system

Nonlinear model

x(t) = Sg(x(t)) + g0

S stoichiometric matrixReaction rate functions g : nonnegative and monotonic

Constant influx vector g0 ≥ 0Dissipative reactions: ∂xi

∂xi< 0

Positive system

Nonlinear model

x(t) = Sg(x(t)) + g0

S stoichiometric matrixReaction rate functions g : nonnegative and monotonicConstant influx vector g0 ≥ 0

Dissipative reactions: ∂xi∂xi

< 0Positive system

Nonlinear model

x(t) = Sg(x(t)) + g0

S stoichiometric matrixReaction rate functions g : nonnegative and monotonicConstant influx vector g0 ≥ 0Dissipative reactions: ∂xi

∂xi< 0

Positive system

Nonlinear model

x(t) = Sg(x(t)) + g0

S stoichiometric matrixReaction rate functions g : nonnegative and monotonicConstant influx vector g0 ≥ 0Dissipative reactions: ∂xi

∂xi< 0

Positive system

Absorb the nonlinear model in a LDI

x(t) = Sg(x(t)) + g0

Idea: g(a, b) − g(a, b) =g(a, b) − g(a, b)

(a− a)︸︷︷︸δa(a,b)

(a− a) +g(a, b) − g(a, b)

(b − b)︸︷︷︸δb(a,b)

(b − b)

Linear Differential Inclusionz.

= x − x , Sg(x) + g0 = 0

z(t) = S [g(z(t) + x)− g(x)] = BD(z(t))Cz(t)

D(z(t)) diagonal positive matrix of partial derivatives

The Jacobian of the original system at the equilibrium has the form

J = BDC

x(t) = Sg(x(t)) + g0

(a− a)︸︷︷︸δa(a,b)

(a− a) +g(a, b) − g(a, b)

(b − b)︸︷︷︸δb(a,b)

(b − b)

= x − x , Sg(x) + g0 = 0

J = BDC

x(t) = Sg(x(t)) + g0

(a− a)︸︷︷︸δa(a,b)

(a− a) +g(a, b) − g(a, b)

(b − b)︸︷︷︸δb(a,b)

(b − b)

= x − x , Sg(x) + g0 = 0

J = BDC

x(t) = Sg(x(t)) + g0

(a− a)︸︷︷︸δa(a,b)

(a− a) +g(a, b) − g(a, b)

(b − b)︸︷︷︸δb(a,b)

(b − b)

= x − x , Sg(x) + g0 = 0

J = BDC

BDC–decomposition: example

z = BDCz ,

D � 0

D = diag{∂gab∂a

,∂gab∂b

,∂gc∂c

,∂gd∂d

,∂gae∂a

,∂gae∂e

−1 −1 1 0 0 −1 −1−1 −1 1 0 0 0 0

1 1 −1 −1 0 0 00 0 0 1 −1 0 00 0 0 0 1 −1 −1

and C =

1 0 0 0 00 1 0 0 00 0 1 0 00 0 1 0 00 0 0 1 01 0 0 0 00 0 0 0 1

Structure: parameter free, no numerical boundsHow can we structurally assess stability (for any Di > 0)?

z = BDCz ,

D � 0

D = diag{∂gab∂a

,∂gab∂b

,∂gc∂c

,∂gd∂d

,∂gae∂a

,∂gae∂e

−1 −1 1 0 0 −1 −1−1 −1 1 0 0 0 0

1 1 −1 −1 0 0 00 0 0 1 −1 0 00 0 0 0 1 −1 −1

and C =

1 0 0 0 00 1 0 0 00 0 1 0 00 0 1 0 00 0 0 1 01 0 0 0 00 0 0 0 1

z = BDCz ,

D � 0

D = diag{∂gab∂a

,∂gab∂b

,∂gc∂c

,∂gd∂d

,∂gae∂a

,∂gae∂e

−1 −1 1 0 0 −1 −1−1 −1 1 0 0 0 0

1 1 −1 −1 0 0 00 0 0 1 −1 0 00 0 0 0 1 −1 −1

and C =

1 0 0 0 00 1 0 0 00 0 1 0 00 0 1 0 00 0 0 1 01 0 0 0 00 0 0 0 1

z = BDCz ,

D � 0

D = diag{∂gab∂a

,∂gab∂b

,∂gc∂c

,∂gd∂d

,∂gae∂a

,∂gae∂e

−1 −1 1 0 0 −1 −1−1 −1 1 0 0 0 0

1 1 −1 −1 0 0 00 0 0 1 −1 0 00 0 0 0 1 −1 −1

and C =

1 0 0 0 00 1 0 0 00 0 1 0 00 0 1 0 00 0 0 1 01 0 0 0 00 0 0 0 1

Structurally assess stability: Lyapunov functions

Quadratic functions

VP(x) = x>Px ,P positive definite matrix

Polyhedral functions

VX (x) = inf{‖w‖1 : Xw = x},X full row rank matrix

conservative

��

non–conservativefor proving robust stability of linear differential inclusionsBrayton & Tong (1980), Molchanov & Pyatnitsky (1986, 1989)

Are quadratic functions suitable for our reaction networks?

Quadratic functions

conservative

��

Quadratic functions

conservative

��

Quadratic functions

conservative

��

Polyhedral Lyapunov functions: a numeric procedure

Idea: associate the Linear Differential Inclusion (1)with a Discrete Difference Inclusion (2)

such that (1) robustly stable iff (2) robustly stablewith the same (weak) polyhedral Lyapunov function

Based on (2), numerical algorithm computes the unit ball ofthe polyhedral Lyapunov function (if any) via set iteration

The procedure converges=⇒ structurally stable

F. Blanchini and G. Giordano, “Piecewise-linear Lyapunov Functions forStructural Stability of Biochemical Networks”, Automatica, vol. 50, n. 10,pp. 2482–2493, 2014

Analysis of motifs I

Monomolecular reactions chain: ∅ a0−⇀ Ag(a)−−⇀ B

g(b)−−⇀ ∅

a = a0 − ga(a)

b = ga(a)− gb(b)

[−α 0α −β

]α = ∂ga/∂a, β = ∂gb/∂b

No P � 0 exists s.t. J>P + PJ � 0 for any choice of α, β > 0

Polyhedral Lyapunov function with X = I2=⇒ Structurally stable, but not quadratically!

Reversible monomolecularreaction: A

g(a)−−⇀↽−−g(b)

BBimolecular reaction: ∅ a0−⇀ A,

∅ b0−⇀ B , A + Bg(a, b)−−−−⇀ ∅

g(b)−−⇀ ∅

a = a0 − ga(a)

b = ga(a)− gb(b)

[−α 0α −β

]α = ∂ga/∂a, β = ∂gb/∂b

g(a)−−⇀↽−−g(b)

∅ b0−⇀ B , A + Bg(a, b)−−−−⇀ ∅

g(b)−−⇀ ∅

a = a0 − ga(a)

b = ga(a)− gb(b)

[−α 0α −β

]α = ∂ga/∂a, β = ∂gb/∂b

g(a)−−⇀↽−−g(b)

∅ b0−⇀ B , A + Bg(a, b)−−−−⇀ ∅

g(b)−−⇀ ∅

a = a0 − ga(a)

b = ga(a)− gb(b)

[−α 0α −β

]α = ∂ga/∂a, β = ∂gb/∂b

g(a)−−⇀↽−−g(b)

∅ b0−⇀ B , A + Bg(a, b)−−−−⇀ ∅

g(b)−−⇀ ∅

a = a0 − ga(a)

b = ga(a)− gb(b)

[−α 0α −β

]α = ∂ga/∂a, β = ∂gb/∂b

g(a)−−⇀↽−−g(b)

∅ b0−⇀ B , A + Bg(a, b)−−−−⇀ ∅

g(b)−−⇀ ∅

a = a0 − ga(a)

b = ga(a)− gb(b)

[−α 0α −β

]α = ∂ga/∂a, β = ∂gb/∂b

g(a)−−⇀↽−−g(b)

Bimolecular reaction: ∅ a0−⇀ A,

∅ b0−⇀ B , A + Bg(a, b)−−−−⇀ ∅

g(b)−−⇀ ∅

a = a0 − ga(a)

b = ga(a)− gb(b)

[−α 0α −β

]α = ∂ga/∂a, β = ∂gb/∂b

g(a)−−⇀↽−−g(b)

∅ b0−⇀ B , A + Bg(a, b)−−−−⇀ ∅

Analysis of motifs II

Bimolecular reversible reaction: A + Bg(a, b)−−−−⇀↽−−−−g(c)

No structural QLF, but PLF with unit ball: −1−0.5

−0.5

Bimolecular–monomolecular reaction chain:∅ a0−⇀ A, ∅ b0−⇀ B , A + B

g(a, b)−−−−⇀ Cg(c)−−⇀ ∅

−0.5

g(a, b)−−−−⇀ Cg(c)−−⇀ ∅

−0.5

g(a, b)−−−−⇀ Cg(c)−−⇀ ∅

−0.5

g(a, b)−−−−⇀ Cg(c)−−⇀ ∅

−0.5

Example

∅ a0−⇀ Ag(a)−−⇀ B + C , B

g(b)−−⇀ ∅, A + Cg(a, c)−−−−⇀ ∅

a = a0 − ga(a)− gac(a, c)

b = ga(a)− gb(b)

c = ga(a)− gac(a, c)

−(α + δ) 0 −γα −β 0

α− δ 0 −γ

α = ∂ga/∂a, β = ∂gb/∂b,γ = ∂gac/∂c , δ = ∂gac/∂a

Not structurally quadratically stable,but structurally stable!

Polyhedral Lyapunov function with unit ball:

−1−0.5

−0.5

Example

∅ a0−⇀ Ag(a)−−⇀ B + C , B

g(b)−−⇀ ∅, A + Cg(a, c)−−−−⇀ ∅

b = ga(a)− gb(b)

−(α + δ) 0 −γα −β 0

α− δ 0 −γ

−1−0.5

−0.5

Example

∅ a0−⇀ Ag(a)−−⇀ B + C , B

g(b)−−⇀ ∅, A + Cg(a, c)−−−−⇀ ∅

b = ga(a)− gb(b)

−(α + δ) 0 −γα −β 0

α− δ 0 −γ

−1−0.5

−0.5

Example

∅ a0−⇀ Ag(a)−−⇀ B + C , B

g(b)−−⇀ ∅, A + Cg(a, c)−−−−⇀ ∅

b = ga(a)− gb(b)

−(α + δ) 0 −γα −β 0

α− δ 0 −γ

−1−0.5

−0.5

Summary

Structural stability of (bio)chemical reaction networks:polyhedral vs. quadratic Lyapunov functions.

Set of basic motifs: structurally stable with polyhedralLyapunov functions, but not quadratically.

For any network that contains one of these motifs,quadratic Lyapunov functions are not suitable.

Yet, for the same network, structural stability may be provedvia polyhedral Lyapunov functions.

Examples are:enzymatic reactions gene expression metabolic networks

Summary

Conclusions

Unfortunately, quadratic Lyapunov functions(well known to be conservative for robust stability of LDIs)cannot help in our setup.

Polyhedral Lyapunov functions: promising tool for assessingstructural stability of biochemical networks.

Ďakujem!

Conclusions

Ďakujem!

Conclusions

Ďakujem!

structural stability of biochemical networks: …

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