modeling and analysis of dynamical distribution...

Modeling and analysis of dynamical distribution networks

Arjan van der Schaft

Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of Groningen, the Netherlands

joint work with Bayu Jayawardhana, Shodhan Rao, Jay Wei

Liege, 22-01-2013

Arjan van der Schaft (Johann Bernoulli Institute for Mathematics and Computer Science University of Groningen, the Netherlands joint work with Bayu Jayawardhana, Shodhan Rao, Jay Wei)Modeling and analysis of dynamical distribution networks Liege, 22-01-2013 1 / 38

Dynamical distribution networks are everywhere

Power networks, supply chains, gas distribution networks, biochemicalreaction networks, · · · .

Network flow theory is concerned with the static throughput aspects: whatis the maximal throughput under flow capacity constraints ?

On the other hand: network dynamics is usually focusing on closedsystems: no inflows and outflows.

Clear need to merge static, optimal, throughput analysis with dynamics.

This talk:(i) Flow control of distribution networks with unknown in/outflows.(ii) Chemical reaction networks motivated by systems biology.


Outline

1 Flow control of distribution networks

2 Chemical reaction networks: basic structure

3 Detailed balanced mass action kinetics

4 Equilibrium analysis and stability

5 Open chemical reaction networks

6 Model reduction of chemical reaction networks

7 Conclusions and outlook


Flow control of distribution networks

Outline










Example: hydraulic network

x1

x2 x3

u1

u2

u3

• xi is fluid storage in i-th reservoir,uj is flow in j-th pipe.

• Dynamical equationsx1x2x3

=

−1 0 11 −1 00 1 −1

u1

u2

u3

• B is incidence matrix of the graph:

bij = −1 if the i-th vertex is tail, or+1 if it is head of the j-th edge

x = Bu



x1

x2 x3

u1

u2

u3

d1 d2

• With inflows and outflowsx1x2x3

=

−1 0 11 −1 00 1 −1

u1

u2

u3

+

01−1

d

• d = d1 = d2 is necessary condition forexistence of equilibrium. In general:

x = Bu + Ed , im E ⊂ im B

Since 1TB = 0 this implies 1TE = 0.



Port-Hamiltonian system defined on the graph

For any Hamiltonian H(x) consider the port-Hamiltonian system

x = Bu, x ∈ Rn (vertices), u ∈ Rm (edges)

y = BT ∂H∂x (x), y ∈ Rm (edges)

satisfying the energy balance

dH

dt= yTu

The control u = −y results in

d

dtH = −∂HT

∂x(x)BBT ∂H

∂x(x) ≤ 0

For H(x) = 12‖x‖

2 this is decentralized control u = −BT x .If the graph is connected then the system will converge to consensus

x1 = x2 = · · · = xnArjan van der Schaft (Johann Bernoulli Institute for Mathematics and Computer Science University of Groningen, the Netherlands joint work with Bayu Jayawardhana, Shodhan Rao, Jay Wei)Modeling and analysis of dynamical distribution networks Liege, 22-01-2013 7 / 38


Inflows and outflows

x = Bu + Ed , x ∈ Rn, u ∈ Rm

y = BT x , y ∈ Rm

Proportional output feedback is not sufficient anymore; we consider aproportional-integral controller structure

xc = yu = −y − xc

The resulting closed loop is[x

xc

]=

[−BBT −B

BT 0

][x

xc

]+

[E

0

]d

which is again a port-Hamiltonian system with total Hamiltonian

1

2(‖x‖2 + ‖xc‖2)



Theorem

Suppose the graph is connected, and im E ⊂ im B. Then the trajectoriesof the closed-loop system for constant d will converge to an element ofthe set

E = {(x , xc) | x = α1, α ∈ R, Bxc = Ed}

with the limit value α uniquely determined by the initial condition x(0).

Proof

Modified Hamiltonian can be used as Lyapunov function, i.e.

V =1

2(‖x‖2 + ‖xc − xc‖2)

d

dtV = −xTBBT x 6 0

Conclusion follows by LaSalle’s invariance principle.



Extension to flow constraints

Stability results can be extended to systems with flow constraints

aj ≤ uj ≤ bj , j = 1, · · · ,m

If aj < 0, bj > 0 no main structural changes.

For aj = 0, bj > 0 (unidirectional flow) there are important changes.

See the papers quoted at the end of the talk, and a paper by J. Wei (to besubmitted).


Chemical reaction networks: basic structure

Outline











Consider a chemical reaction network with m chemical species, amongwhich r chemical reactions take place.The dynamics of the vector x ∈ Rm

+ of concentrations is given by thebalance laws

x = Sv ,

where S is an m × r matrix, called the stoichiometric matrix, consisting of(positive and negative) integer elements. For example, for the networkconsisting of two (reversible) reactions

X1 + 2X2 X3 2X1 + X2

we have

S =

−1 2−2 11 −1

The vector v ∈ Rr are called the reaction fluxes, or reaction rates,whenever they are expressed as a function v(x).



The graph of complexes

Define the set of complexes as the union of all the different left- andright-hand sides (substrates and products) of all the reactions in thenetwork.The expression of the c complexes in terms of the m chemical speciesdefines an m × c matrix Z called the complex stoichiometric matrix.The ρ-th column of Z is the expression of the ρ-th complex in the mchemical species.For the network X1 + 2X2 X3 2X1 + X2 we have

Z =

1 0 22 0 10 1 0

All elements of the matrix Z are non-negative integers.



The complexes define the vertices of a directed graph, with edgescorresponding to the reactions. The resulting graph is called the graph ofcomplexes. In biochemical reaction networks c , r are typically large.As before the complex graph is specified by its incidence matrix B: anc × r matrix with (ρ, j)-th element equal to −1 if vertex ρ is the tail vertexof edge j and 1 if vertex ρ is the head vertex of edge j , while 0 otherwise.In the previous case

B =

−1 01 −10 1

Basic relation

S = ZB

Hence the dynamics x = Sv can be also written as

x = ZBv ,

with Bv belonging to the space of complexes Rc .


Detailed balanced mass action kinetics

Outline










The most basic model for specifying the reaction rates v(x) is mass actionkinetics. Consider the single reaction

X1 + 2X2 X3,

involving the three chemical species X1,X2,X3 with concentrationsx1, x2, x3. In mass action kinetics the reaction is considered to be acombination of the forward reaction X1 + X2 ⇀ X3 with forward rateequation v+(x1, x2) = k+x1x2

2 and the reverse reaction X1 + X2 ↽ X3,with rate equation v−(x3) = k−x3. The constants k+, k− are called theforward and reverse reaction constants.The net reaction rate given by mass action kinetics is thus

v(x1, x2, x3) = k+x1x22 − k−x3



In general, the mass action reaction rate of the j-th reaction, from asubstrate complex Sj to a product complex Pj , is given as

vj(x) = k+j

m∏i=1

xZiSji − k−j

m∏i=1

xZiPji

where Ziρ is the (i , ρ)-th element of the complex stoichiometric matrix Z ,and k+

j , k−j ≥ 0 are the forward/reverse reaction constants of the j-th

reaction.Crucial observation: this can be also written as

vj(x) = k+j exp

(ZTSjLn(x)

)− k−j exp

(ZTPjLn(x)

)where ZSj and ZPj

are the columns of the complex stoichiometric matrix Zcorresponding to the substrate and the product complex of the j-threaction.(The mapping Ln : Rc

+ → Rc is the element-wise natural logarithm.)



Definition

A vector of concentrations x∗ ∈ Rm+ is called a thermodynamic equilibrium

ifv(x∗) = 0

Clearly, any thermodynamic equilibrium is an equilibrium of x = Sv(x),i.e., Sv(x∗) = 0, but not necessarily the other way around (since in generalS = ZB is not injective).

Basic paradigm from thermodynamics:

every well-defined chemical reaction network has a thermodynamicequilibrium

This corresponds to microscopic reversibility.



Necessary and sufficient conditions for the existence of a thermodynamicequilibrium are usually referred to as the Wegscheider conditions.Consider the j-th reaction from substrate Sj to product Pj , described bymass action kinetics

vj(x) = k+j exp

(ZTSjLn(x)

)− k−j exp

(ZTPjLn(x)

)Then v(x∗) = 0 if and only if (detailed balance equations)

k+j exp

(ZTSjLn(x∗)

)= k−j exp

(ZTPjLn(x∗)

), j = 1, · · · , r

Define the equilibrium constant K eqj of the j-th reaction as

K eqj :=

k+j

k−j

Then the detailed balance equations are equivalent to

K eqj = exp

(ZTPjLn(x∗)− ZT

SjLn(x∗)), j = 1, · · · , r



Collecting all reactions this amounts to the vector equation

K eq = Exp(

BTZTLn(x∗))

= Exp(

STLn(x∗)),

where K eq is the r -dimensional vector with j-th element K eqj , j = 1, · · · , r ,

and Exp : Rc → Rc+ is the element-wise exponential function.

Proposition

There exists a thermodynamic equilibrium x∗ ∈ Rm+ if and only if

k+j > 0, k−j > 0, for all j = 1, · · · , r , and furthermore

Ln (K eq) ∈ im ST

Hence all reactions should be reversible, and in case ST is not surjective,then the forward and reverse reaction constants should satisfy acompatibility condition ! Such a network is called detailed balanced.It also follows that the set of all thermodynamic equilibria is equal to

E := {x∗∗ ∈ Rm+ | STLn(x∗∗) = STLn(x∗)}



Define for each reaction the (reaction) conductance

κj(x∗) := k+j exp

(ZTSjLn(x∗)

)= k−j exp

(ZTPjLn(x∗)

), j = 1, · · · , r

Then the reaction rate of the j-th reaction can be rewritten as

vj(x) = κj(x∗)[exp

(ZTSjLn

( x

x∗

))− exp

(ZTPjLn( x

x∗

))],

where for any vectors x , z ∈ Rm the quotient vector xz ∈ Rm is defined

elementwise.



Define the r × r diagonal matrix of conductances as

K(x∗) := diag(κ1(x∗), · · · , κr (x∗)

)It follows that the mass action reaction rate vector of a balanced reactionnetwork can be written as

v(x) = −K(x∗)BTExp(

ZTLn( x

x∗

)),

and thus the dynamics of a balanced reaction network takes the final form

x = −ZBK(x∗)BTExp(

ZTLn( x

x∗

)), K(x∗) > 0



Discussion

The matrix L := BK(x∗)BT defines a weighted Laplacian matrix for thegraph of complexes, with weights being the constants κ1(x∗), · · · ,κr (x∗).

Structure of the equations is very similar to flow control distributionsystems under the proportional output feedback u = −K(x∗)y . Thus weexpect ’convergence to consensus’.

The Laplacian matrix L is an important tool for the analysis of the graph(algebraic graph theory). E.g., it will be key to the model reductionprocedure discussed later on.

K(x∗), and thus the Laplacian matrix L, is in principle dependent on thechoice of the thermodynamic equilibrium x∗. However, this dependence isminor: for a connected complex graph the matrix K(x∗) is, up to apositive multiplicative factor, independent of the choice of thethermodynamic equilibrium x∗.



Thermodynamic interpretation

We may interpret

µ(x) := RTLn( x

x∗

)= −RTLn(x∗) + RTLn(x)

as the vector of chemical potentials. Furthermore, γ(x) := ZTµ(x), thecomplex thermodynamical affinity, acts as a ‘driving force’ for thereactions. Define the Gibbs’ free energy as

G (x) = RTxTLn( x

x∗

)+ RT (x∗ − x)T 1m,

Then ∂G∂x (x) = RTLn

(xx∗

)= µ(x) and

x = −ZBK(x∗)BTExp

(ZT ∂G

∂x(x)

), µ(x) =

∂G

∂x(x)

Note that (RT )Ln(K eq) = (RT )STLn(x∗) = −BTγo , and thus theequilibrium constants correspond to differences of the reference complexthermodynamical affinities of the substrate and product complexes.



Summarizing diagram

All variables and relations are summarized in the following diagram

v ∈ R = Rr B−→ y ∈ C = Rc Z−→ x ∈M = Rm

K(x∗) | | G (x)

v∗ ∈ R∗ ←−BT

γ ∈ C∗ ←−ZT

µ ∈M∗

�Exp

where µ(x) = ∂G∂x (x).

Compared to other cases of physical network modeling a complication inthe diagram is the map Exp : C∗ → C∗, which introduces a discrepancybetween v∗ and α := −BTγ = −STµ.


Equilibrium analysis and stability

Outline










Theorem

Consider a balanced chemical reaction network x = Sv(x) = ZBv(x)governed by mass action kinetics, with thermodynamic equilibrium x∗.Then all equilibria are thermodynamic equilibria, i.e., in

E := {x∗∗ ∈ Rm+ | STLn(x∗∗) = STLn(x∗)}



Proof.

Denote for the j-th reaction the substrate complex by Sj and the productcomplex by Pj . Let ZSj and ZPj

denote the columns of the complexstoichiometric matrix Z , corresponding to substrate complex Sj andproduct complex Pj . Recall

µ(x) = Ln( x

x∗

), γ(x) := ZTµ(x), γSj (x) = ZT

Sjµ(x), γPj(x) = ZT

Pjµ(x)

Suppose x∗∗ is an equilibrium, i.e., ZBK(x∗)BTExp(ZTµ(x∗∗)

)= 0.

ThenµT (x∗∗)ZBK(x∗)BTExp

(ZTµ(x∗∗)

)= 0

Denoting the columns of B by b1, · · · , br , we haveL = BK(x∗)BT =

∑rj=1 κj(x∗)bjb

Tj , while

µT (x∗∗)Zbj = µT (x∗∗)(ZPj− ZSj

)= γTPj

(x∗∗)− γTSj (x∗∗),

bTj Exp

(ZTµ(x∗∗)

)= exp

(γTPj

(x∗∗))− exp

(γTSj (x∗∗)

)Arjan van der Schaft (Johann Bernoulli Institute for Mathematics and Computer Science University of Groningen, the Netherlands joint work with Bayu Jayawardhana, Shodhan Rao, Jay Wei)Modeling and analysis of dynamical distribution networks Liege, 22-01-2013 28 / 38


Proof.

It follows that

0 = µT (x∗∗)ZBK(x∗)BTExp(ZTµ(x∗∗)

)=∑r

j=1

[γPj

(x∗∗)− γSj (x∗∗)] [

exp(γPj

(x∗∗))− exp

(γSj (x∗∗)

)]κj(x∗)

Since the exponential function is strictly increasing andκj(x∗) > 0, j = 1, · · · , r , this implies that all terms in the summation arezero and thus

γPj(x∗∗) = γSj (x∗∗), j = 1, · · · , r

exp(γPj

(x∗∗))

= exp(γSj (x∗∗)

), j = 1, · · · , r

which implies

BTExp(γ(x∗∗)) = BTExp(ZTµ(x∗∗)) = 0, and thus v(x∗∗) = 0



Theorem

Consider a balanced mass action reaction network. For every initialcondition x0 ∈ Rm

+ there exists a unique point x∞ ∈ E withx∞ − x0 ∈ im S.If the network is persistent, the species concentration vector x startingfrom x(0) = x0 converges for t →∞ to x∞.



Proof.

There exists a unique µ ∈ ker ST such that x∗ · Exp(µ)− x0 ∈ im S .Define x∞ := x∗ · Exp(µ) ∈ Rm

+.

It follows that 0 = STµ = STLn(x∞x∗

), that is, x∞ ∈ E . Furthermore, the

function G satisfies

G (x∗) = 0, G (x) > 0, ∀x 6= x∗,

while G (x) := ∂TG∂x (x)Sv(x) = dG

dt (x) satisfies

G (x) ≤ 0 ∀x ∈ Rm+, G (x) = 0 if and only if x ∈ E .

The convergence to E is equivalent to BTγ(x(t))→ 0 for t →∞.Thus the elements of γ belonging to the same connected component ofthe complex graph converge to the same value (which is determined byx0); similar to standard consensus algorithms.


Open chemical reaction networks

Outline









Open chemical reaction networks

Open balanced chemical reaction networks are given by

x = −ZBK(x∗)BTExp(ZTLn

(xx∗

))+ Sbvb,

µb = STb Ln

(xx∗

),

which define an input-state-output system with inputs vb (boundaryfluxes) and outputs µb (boundary potentials). It follows that

d

dtG (x) = −γT (x)BK(x∗)BTExp(γ(x)) + µTb vb

thus showing the passivity property

d

dtG ≤ µTb vb

What can we say about steady state behavior for constant vb (similar toconstant d for flow control distribution networks) ?


Model reduction of chemical reaction networks

Outline










Reduction of weighted directed graphs

Proposition

Consider a directed graph with vertex set V and with weighted Laplacianmatrix L = BK(x∗)BT .For any subset of vertices Vr ⊂ V the Schur complement of L with respectto the indices corresponding to Vr is well-defined, and is the weightedLaplacian matrix L = BK(x∗)BT of a reduced directed graph withincidence matrix B, whose vertex set is equal to V − Vr .



Consider a balanced reaction network

Σ : x = −ZBK(x∗)BTExp(

ZTLn( x

x∗

))Reduction is performed by deleting certain intermediate complexes in thecomplex graph, resulting in a reduced reaction network

Σ : x = −Z BK(x∗)BTExp(

ZTLn( x

x∗

)), K(x∗) > 0

Σ is again a balanced chemical reaction network governed by mass actionkinetics, with a reduced number of complexes. The thermodynamicequilibrium x∗ of the original reaction network Σ is also a thermodynamicequilibrium of the reduced network Σ.


Conclusions and outlook

Outline











• Distribution networks are everywhere: clear need to combine networkstructure, dynamics, and throughput analysis.

• Current work is focusing on control under flow capacity constraints.

• Structured description of chemical reaction networks: dynamics issimilar to mass-damper systems with nonlinear dampers.

• Framework can be extended from mass action kinetics toMichaelis-Menten kinetics (enzymatic reactions). Model reductionapproach has been applied to a glycolysis network.

• Certainly for biochemical reaction networks the study of networkswith external fluxes vb is essential: many open questions !

• We started to look at feedback loops from certain metabolites to theenzymes, corresponding e.g. to gene activity.

• Papers can be found on my homepage: www.math.rug.nl/˜arjan


modeling and analysis of dynamical distribution...

Documents