control and system theory for biochemical reaction networks

Modeling Rational positive systems Control of biochemical networks System reduction System identification Further research

Control and System Theoryfor Biochemical Reaction Networks

Jan H. van SchuppenHanna Hardin (VU/CWI), Jana Nemcova (CWI),

Ilona Verburg (VU)

Centrum voor Wiskunde en Informatica (CWI), Amsterdam

20 October 2006Workshop Stochastic SystemsSZTAKI, Budapest, Hungary


Outline

1 Modeling

2 Rational positive systems

3 Control of biochemical networks

4 System reduction

5 System identification

6 Further research


Modeling of biochemical reaction networks

Example Glycolysis in Trypanosoma brucei

Trypanosoma brucei (Tb). Unicellular eukaryote (with nucleus).Parasite in humans and other mammals. Lives in blood andtissue.

Subspieces causes the African Sleep Disease. 200.000 newinfections a year. Lethal unless treated. Damage to lifestock.

Need for medicines. Existing drugs have severe side effects.

Research

Paul Michels and Fred Opperdoes (Institute of CellularPathology, Universite Catholique de Louvain, Brussels, Belgium).

Barbara M. Bakker and Hans V. Westerhoff (Department ofBiology, Vrije Universiteit, Amsterdam, The Netherlands).


Example. Glycolysis in T. brucei

T. brucei gets energy from host in the form of glucose.Glucose is consumed only in the process of glycolysis.

Glycolysis :From glucose (sugar) to ATP, to pyruvate (90%), and toglycerol (10%).

Uniquely in this organism, glycolys is largely performed inorganelles called glycosomes .

Search for medicines directed at enzymes/reactionsthat limit the release of energy.

Next page: Figure of T. brucei from (B. Bakker (1998), p. 8).Original source I. Coppens (IPC, Brussels).Sources for second figure: Source 1: (I. Coppens (ICP, Brussels))Source 2: (B.M. Bakker, Ph.D. thesis, VU, Amsterdam, 1998, p. 9).


Glycolysis in T. brucei


Glycolysis in T. brucei

Glycolysis = splitting of sugar (glucose) into C3 molecules.In human red-blood cells,

C6H12O6 + ADP + P → 2C3H6O3(lactic acid) + ATP.

ATP = Adenosine Triphosphate used elsewhere in cell for activities.Conversion ATP → ADP + phosphate, delivers the energy.Reactions For example,

Glex ↔ Glcc ↔ Glcg

Glcg + ATPgHK→ Glc − 6− Pg + ADPg

Glc − 6− Pg PGI↔ Fru − 6− Pg , . . .

19 reversible and 4 irreversible reactions.


Biochemical system for glycolysis inTrypanosoma brucei

States represent concentrations in the glycosome, cytosol,mytochondria, external environment, overall, and the average.

x1 = [GLC]g , x2 = [Glc]ex , x3 = [Glc − 6− P]g ,

x4 = [Fru − 6− P]g , x5 = [Fru − 1, 6− BP]g , x6 = [ATP]g ,

x7 = [ADP]g , x8 = [AMP]g , x9 = [DHAP]g , . . .

Enzymes and corresponding input components:

u1 transport of glucose through the plasma and

through the glycosome membrane,

u2 HK, u3 PGI,, u4 PFK,

u5 ALD, u6 TIM, u7 GAPDH, . . .


Biochemical system for glycolysis in T. brucei

Notation.

vi = riui , ri rate function ui enzyme concentration .

r1(x) = c1,1c1,2(x2 − x1)

1 + c1,2x1 + c1,2x2 + c1,3c21,2x1x2

,

Glcg + ATPgHK→ Glc − 6− Pg + ADPg ,

r2(x) = c2,1c2,2c2,3x1x6

(1 + c2,2x6 + c2,4x7)(1 + c2,3x1),

r7(x) = c7,1[c7,3x10c7,4x24 − c7,2c7,5c7,6x12c8x25]

(1 + c7,3x10 + c7,5x12)(1 + c7,4x24 + c7,6x25),

etc.



Differential equation,

d [Glc]indt

=vglucosetransport − vHK

Vtot= ctot,1(r1(x)u1 − r2(x)u2);

x(t) =m∑

i=1

m∑j=1

(bi − bj)ri,j(x(t))ui,j(t),

=m∑

k=1

(b+k − b−k )rk (x(t))uk (t), (k ∼ (i , j)),

= BDiag(r(x(t)))u(t), x(t0) = x0.

B called the stoichiometric matrix(stoikheion = elements (from Greek)).



Moiety conservation relations (there exist five).

[ATP]g + [ADP]g + [AMP]g = c1, ⇔ x6 + x7 + x8 = c1.

[NADH]g + [NAD]+g = c3, ⇔ x24 + x25 = c3.

Pools . If a reaction is assumed to be in equilibrium then the substrateand the products are treated as a single metabolic pool. There arefive pools.

[hexose− P]g = [Glc − 6− P]g + [Fru − 6− P]g

⇔ x30 = x3 + x4, etc.

Fast dynamics Five relations for fast reactions.

Glc − 6− PgPGI↔ Fru − 6− Pg x3

u3↔ x4x4

x3= ceq,3.



Differential-algebraic rational positive system.Algebraic operations lead to the nonlinear system.Def. Nonlinear positive system of Trypanosoma brucei

x(t) = BDiag(r(x(t))u(t), x(t0) = x0,

z(t) = HDiag(r(x(t))u(t), outflow rate.

10 ordinary differential equations, 12 enzymes.(I. Verburg, 2006, pp. 130-133.)Questions

Is system positive? Yes!

Steady states? One unique steady state, apparently! No proofyet.

Graph? Two irreducible components (sizes 9,1).


Further research - Modeling of biochemicalreaction networks

Types of networks

Metabolic networks.

Signal transduction networks.

Gene networks.

Model for all reactions in a cell

Hierarchical model: genetic level, signal level, metabolic level.

Interations between levels.

Realistic and mathematically treatable small models?

Main problem: How to derive realistic models of low complexity?Numbers are huge:Humans: 30.000 genes, 15.000 enzymes/reactions, 10.000 conc.E. coli, 4000, 2000, 1000. Yeast 6000.


Outline

1 Modeling



4 System reduction


6 Further research


Rational positive system

Def. Rational positive system for cell reaction network

x(t) = BDiag(r(x(t), xex(t)))u(t), x(t0) = x0,

z(t) = HDiag(r(x(t), xex(t)))u(t), outflow rate,

y(t) = Cx(t), output ,

T = [t0,∞), n, m ∈ Z+, nex , nz ∈ N,

X = Rn+, Xex = Rnex

+ , U = Rm+,

rj(x , xex) =p+

j (x , xex)

qj(x , xex)−

p−j (x , xex)

qj(x , xex),

(p+j /qj), (p

−j /qj) ∈ R+,s(x , xex), ∀j ∈ Zm,

x(t) =m∑

j=1

(B+j − B−j )

[p+

j (x(t), xex)

qj(x(t), xex)−

p−j (x(t), xex)

qj(x(t), xex)

]uj(t).


Rational positive system

Conditions imposed:1 (Relative primeness)∀j ∈ Zm, (p+

j , qj), (p−j , qj) relatively prime in R+[x ].

2 (Forward invariance) ∀i ∈ Zn, j ∈ Zm,

xi = 0 ∧ B+i,j − B−i,j > 0 ⇒ p−j (x) = 0,

xi = 0 ∧ B+i,j − B−i,j < 0 ⇒ p+

j (x) = 0.

Hi,k > 0 ⇒ p−j = 0,∀k ∈ Zm.

3 (Linear independence) {rj(.) ∈ R[x ], j ∈ Zm}.4 (Existence and uniqueness solution)

For all T = [t0,∞), x0 ∈ Rn, xex , u,there exists an unique solution of the ODE.


Further research - System theory of biochemicalsystems

Biochemical system for a reaction network.

x(t) = BDiag(r(x(t)))u(t), x(t0) = x0,

z(t) = HDiag(r(x(t)))u(t), rates of flows from/to environment,

y(t) = Cx(t), measurements.

Issues

Reversible and irreversible reactions with environment.

From system equation follow the conservation relationsSx(t) = c.

Decomposition into irreducible subsystems.

Classification of types of irreducible subsystems.

Controllability and observability.

Realization theory.


Outline

1 Modeling



4 System reduction


6 Further research


Problem. Control of biochemical reactionnetworks

How does the cell regulate its biochemical reaction network?

x-

--

u

xex

z

Subproblems

(a) Which enzymes concentration(s) to put to zero so as to achievea zero outflow for one of its controlled outputs? Motivated byrational drug design.

(b) How does the cell determine enzymes to be increased to achievethe control objective of a higher value for one of its functions?

(c) How does the biochemical reaction network react against abruptchanges in its food supply?


Control for network-based drug design

Needs of cell biologists for control:1 Understanding role of feedback in cell reaction networks.2 Research for medical drugs.

Network-based drug design. Phases:1 Model the reaction network of the cell for the phenomenon.2 Select reactions and corresponding enzymes to inhibit.3 Select chemical compounds to attach to active site.4 Study side effects.


Drug effect by inhibition

Control actuation based on inhibition of a chemical reaction

Biochemical reaction catalyzed by an enzyme.Enzyme is a large molecule.

Active site is the location on the enzyme where the substratesbinds to the enzyme and the reaction occurs.

Chemical substances introduced into the cell via a drug mayattach to active site. Substance will stay there for a long time.

Inhibition of biochemical reaction:regular reaction cannot take place because active site isoccupied by another chemical substance.


Approaches to control of reaction networks

Approach of biochemists Approach to zeroing output for drugdesign:(1) Simulation. (2) Metabolic control theory (sensitivity analysis).Approaches of control theory

1 Random search - One, two, or more reactions inhibited:Select ui = 0 for i ∈ I ⊂ Zm. Computation of steady states.

2 Search for a minimal colored cut set in the graph of the reactions,see below.

3 Determine control law or input signal such that the zerodynamics of the controlled output is globally attractive.

4 Decoupling for control of particular outflows.

RemarkMetabolic control theory developed by H. Kacser and J.A. Burns(1973), R. Heinrich and T.A. Rapoport (1975), and many others.


Problem. Control for drug design

Determine the minimal subset of enzymes to inhibit so as toachieve a zero outflow rate Consider k ∈ Znz . Determine the subset

J0 ⊆ Zm, such that,

if {uj = 0, ∀j ∈ J0} then zk = 0.

Def. Define an edged-colored directed graph(Nodes, edges) = ((states, outflows, inflows), reactions).Color corresponds to a particular enzyme-catalyzed reaction.Consider a particular inflow and a particular outflow and theircorresponding nodes v0, v1 ∈ V .A colored cut set is a set of edges which, when these edges aredeleted from the network, results in a graph for which there does notexist a path from vertex v0 to vertex v1.Problem The minimal colored cut set problem is to determine acolored cut set with the minimal number of colors.


Case study of control for drug design:Tryapanosoma brucei

Graph based on reaction network.If one or more enzymes are inhibited does there then exist a pathfrom glucose to the flow of pyruvate?Aerobic case . Production of ATP in glycosome stopped if one of thefollowing subsets of enzymes are inhibited:

one enzym: u1, . . . , u5, u7, u8;

two enzymes: -.

Simulations for u1 = 0 and for u4 = 0 show this.Anaerobic case . Production of ATP in glycosome stopped if one ofthe following subset of enzymes are inhibited:

one enzym: u1, . . . , u5;

two enzymes: (u6, u7) and (u8, u9).

Simulations for u1 = 0 and for (u6, u7) = (0, 0) show ATP productionstopped.


Further research - Control of metabolic networks

Problem Control of rational positive systems

x(t) = BDiag(r(x(t)))u(t), x(t0) = x0,

z(t) = HDiag(r(x(t)))u(t).

Determine a control law g : X → U such that the input functionu(t) = g(x(t)) achieves the control objectives.For example, the control objective of prespecified flow rate zs ∈ Rpz

+ .Issues

Motivated by regulation of metabolic networks by the geneticnetwork.

How does the genetic network do this?

Regulation of all input components or a small subset?

Controllability of rational positive systems needed.


Outline

1 Modeling



4 System reduction


6 Further research


Problem. System reduction

Consider the rational positive system(the high-order system)

x(t) = BDiag(r(x(t), xex(t)))u(t), x(t0) = x0,

z(t) = HDiag(r(x(t), xex(t)))u(t), x(t) ∈ Rn1+ , n1 ∈ Z+.

Determine an algorithm which produces a second rational positivesystem of lower order than the original one

xr (t) = Br Diag(rr (xr (t), xex(t)))u(t), xr (t0) = xr ,0,

zr (t) = Hr Diag(rr (xr (t), xex(t)))u(t), xr (t) ∈ Rnr+ , nr � n1,

such that the following approximation criterion is as small as possible,

‖z(u, xex)− zr (u, xex)‖p, ∀u ∈ F (T , U), ∀xex ∈ F (T , Xex).


Remarks on system reduction problem

High-order system is a positive system.

Should reduced-order system be positive? Preferably!

Approximation criterion?

Emphasis on algorithms.

Theoretical aim of optimality of algorithms for approximationcriterion.

System reduction for biochemical reaction networks differs fromthat for control systems due to reversibility of reactions.

Preservation of conservation of mass or dissipativity of rationalpositive systems in system reduction.

Motivation Investigation of large biochemical networks.System identification.


Approaches to system reduction

1 Dynamical system theory. Time-scale decomposition. Singularperturbation. Tikhonov’s theory.

2 System reduction of the input-output operator.Developed well for linear systems and partly for nonlinearsystems. Book A.C. Antoulas (2005).Proceedings P. Benner, V. Mehrmann (2005).

3 System reduction of positive systems.Algebraic decomposition into irreducible subsystems.System reduction for irreducible subsystems.

4 System theory and system identification. Algorithms analogousto algorithm of subspace identification.

Research project System reduction of biochemical reaction networks.Hanna Hardin, Hans V. Westerhoff, JHvS (Vrije Universiteit and CWI).


Specific approaches for system reduction ofpositive systems

1 System reduction from a nonlinear positive systemby linearization at a steady state,balancing of the linear systems, followed by truncation.Hanna Hardin and JHvS (2006). See below.

2 System reduction from a nonlinear positive system by nonlinearbalancing and truncation. (J. Scherpen (2004))

3 System reduction from a linear positive system to areduced-order linear positive system.Not directly relevant for biochemical reaction networks but anintermediate step.

4 System reduction from a rational positive system to a rationalpositive system of lower order.


Case study of system reduction:Glycolysis in yeast

Model (B. Teusink etal(2000)).

Insidethe cell

the cellOutside

Succinate

Dihydroxyacetone phosphate

Glucose

Cell membrane

Glycogen Trehalose

Enzymatic transportthrough cell membrane

Glucose (x1)

Glucose−6−phosphate (x2)

Fructose−6−phosphate (x3)

Fructose−1,6−biphosphate (x4)

x5

1,3−Biphosphoglycerate (x6)

3−Phosphoglycerate (x7)

2−Phosphoglycerate (x8)

Phosphoenolpyruvate (x9)

Pyruvate (x10)

Acetaldehyde (x11)

Ethanol (x13)

Glycerol (x12)

Glyceraldehyde 3−phosphate



Baker’s yeast (Sacccharomyces cerevisiae). To be approximatedsystem: Rational positive system for glycolysis in yeast.Input is glucose concentration outside the cell,output is concentreation (flow) of pyruvate.

System reduction algorithm:1 Computation of steady state,2 linearization of nonlinear system at steady state,3 balancing, and4 truncation.



Approximant system

0 = f (xs, us), n = 13, p = 1,

x(t) = x(t)− xs,

xr (t) = Ar xr (t) + Br u(t), xr (t) = xr ,0,

zr (t) = Cr xr (t), nr = 3, pr = 1,

Ar =

−0.2.5707 7.4384 3.1752−7.4384 −17.5145 −20.4705

3.1750 20.4697 −67.3467

, Br =

−0.1862−0.2161

0.1167

,

Cr =(−0.1862 −0.2161 0.1167

).



Stepresponse of the pyruvate concentration after a change in theglucose concentration of (a) 10 % and (b) 50%.

0 0.5 1 1.5 2time@minD

8.53

8.54

8.55

8.56

8.57

8.58

conc

@mmol�LD

HaL

0 0.5 1 1.5 2time@minD

8.55

8.6

8.65

8.7

8.75

8.8

conc

@mmol�LD

HbL



Reasons for reduction?

Conservation relations. There are none, verified by computation.

Time-scale decomposition (fast-slow dynamics). Linearizedsystem at steady state has as eigenvalues of the system matrix:

−2.72366, −6.49762, −21.6752, −80.3423, . . . , −1.21324∗106.

Input-output operator. Hankel singular values are

0.0067, 0.0013, 0.0001, 0, . . . , 0.


Further research - System reduction

Issues

Purpose of reduced systems.

Preservation of conservation or dissipativity in system reduction.

Examples of large networks of nonlinear positive systems.

Decomposition of large networks.

Interaction of reduced systems.

Algorithms for system reduction of rational positive systems.

Optimality of algorithms.


Outline

1 Modeling



4 System reduction


6 Further research


Further research - System identification ofbiochemical systems

Remarks

Computational models require numerical values of parameters.

Current practice for obtaining paramater values.

Measurements of cell concentrations require extensive work.

Approach1 Realization theory of rational positive systems (CWI).2 Identifiability.3 Approximation algorithms.

Analogy with subspace identification of Gaussian systems.4 System reduction.


Outline

1 Modeling



4 System reduction


6 Further research


Conclusions so far

Detailed biochemical modeling possible and necessary forcontrol and system theory.

Control for network-based drug design practically possible.Multi-target drugs necessary.

Control of metabolic networks . Problem formulations. Needsfurther research.

System reduction useful.Case of rational positive systems requires attention.

System identification requires attention.Use biochemical modeling, system reduction, and identificationof reduced model.


Further research - Systems biology

Experience with a large variety of biochemical networks.

Investigation of very large networks.Best done with EU-wide teams of researchers.

Genetic networks and their interaction with signal and metabolicnetworks.

Control for networked-based drug design.

Control for biotechnology.


Further research - Control and system theory

System theory of biochemical systems. Reversible reactions.

Control theory. Control of networks.Applications: Drug design and biotechnology.

System reduction. Irreducible rational subsystems.

System identification.

Observers for rational positive systems.

Realization theory for rational positive systems. (CWI).

Software package for modeling, system reduction, systemidentification, observers, control, and realization.


Acknowledgements

Barbara M. Bakker, Hanna Hardin, Hans V. Westerhoff(VU.SystemsBiology).

Ilona W.M. Verburg, Andre C.M. Ran (VU.Math).

Jana Nemcova (CWI).

Financial support in part

CWI (Amsterdam, The Netherlands).

Vrije Universiteit (Amsterdam).

NWO, including the Computational Life Sciences Program.

The End!

control and system theory for biochemical reaction networks

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