A set of interconnected chemical reaction is commonly referred to as a chemical reaction network .A chemical reaction network (CRN) is a set of chemical reaction interconnected by reactant or product and denoted by a finite set ℝ :={Ri , i=1,2.....r} where r is number of reaction. Reactant and product of the reactions are called species � :={Sj ,j=1,2,...N} where N is number of species. A chemical reaction network can represent by a set of non-linear differential equation.These differential equation can be formulate using general mass action kinetics. The law of mass action is: Reaction rates (at constant temperature) are proportional to product of concentration. The proportionality factor (rate constant ) is depend upon temperature , reactant etc. Example:

A transformation k

X Y→ gives differential eqation like

[ ] [ ]

[ ] [ ]

d X k Xdt

d Y k Xdt

= −

=

Now we begin with general formulation of kinetic equation that describes the chemical reaction using mass action kinetics. Individual chemical reaction is denoted as: ℝ i = ij j ij jj S j S

S Sα β∈ ∈

=∑ ∑

Where αij , βij are nonnegative integer called the Stoichiometric coefficient.

It is convenient to arrange Stoichiometric into a N×nr matrix Γ, called stoichiometric matrix,which is defined as follows:

[Г]ji= βij-αij ∀ i∈ ℝ and j∈ �

Example:

1 3

24 6

5

R R

RR R

R

E S ES E P

F P FP S F

+ → +

+ → +

�

�

Species set:={E,S,ES,P,F,FP} and reaction set:={R1,R2,R3,R4,R5,R6}

Species vector=

ESESPFFP

Reactant coefficient matrix[α]=

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

Product coefficient matrix[β]=

0 1 1 0 0 00 1 0 0 0 11 0 0 0 0 00 0 1 0 1 10 0 0 0 1 00 0 0 1 0 0

Stoichiometric matrix[Г]=[β-α]T

Г =

1 1 1 0 0 01 1 0 0 0 1

1 1 1 0 0 00 0 1 1 1 00 0 0 1 1 10 0 0 1 1 1

− − − − −

− −

Now the differential equation associated with CRN is given as follows:

dSj/dt =ГR(S) , ∀ Sj∈�

where R(S) is a vector that describes how the state of the network evolves with times for a given CRN.

In the mass-action kinetics it assume that ,

Rj(S)= kj1

ijN

ii

Sα

=∏ ∀ j= 1,....nr ,where nr is the number of reaction.

R(s):=

1( )2( )

.

.

. ( )r

R SR S

Rn S

With this convention the system differential equation associated to the CRN is given as follows:

dS/dt= ГR(S)

Deficiency theoremà

Terminology: Basis vector: With each species we associate a vector ei, where {e1,. ..., es} is the standard basis for RS,

e1=

100..0

e2=

010..0

........es=

00..01

Complex: Complex of a network is the object that appears before and after the reaction arrow. Stoichiometric subspace: The reaction vectors span is a linear subspace � ∊ ℝN which is called the stoichiometric subspace. Reaction vector: Each column of Stoichiometric matrix treats as a reaction vector. . Stoichiometric compatibility class(c): Two composition c and c', for a reaction network are stoichiometrically compatible if c’-c lies is stoichiometric subspace for corresponding reaction network. Linkage class(l ): linkage class of a network is the set of complexes which are linked to each other. Reversibility and weak reversibility: A reaction network is reversible if each reaction accompanied by its reverse reaction. We shall say a reaction network is weakly reversible if whenever there exists a directed pathway (consisting one or more reaction arrows) pointing from one complex to other, there exists another directed pathway that pointing from second complex to the first one. Strongly linked:Two complexes in a network called strongly linked if there exists a directed arrow pathway pointing from one complex to the other and a directed arrow pathway pointing from the second complex back to the first.

N.B: Every complex is strongly linked to itself. Strong linkage class: A set of complexes such that every pair of complexes within the set is strongly linked, and no complex in the set is strongly linked to a complex that is not in the set. Terminal strong linkage class : A strong linkage class is terminal if there is no exit from it along a directed arrow pathway.

Deficiency of a network:

δ:= c-l -ρ where c= number of complex

l = number of linkage class

ρ= rank of Stoichiometric matrix

Multistability of CRN:

Chemical reaction networks have ability to reach multiple equilibrium point (multistability) for different initial condition. That is there will be a change in equilibrium point if there is change in initial condition.

Deficiency zero theorem :à

1. If the given chemical reaction network is weakly reversible and

2. The given CRN has zero deficiency.

Then, for any choice of rate constants, within each positive stoichiometric compatibility class there is one equilibrium point, and that equilibrium is locally asymptotically stable relative to its compatibility class.

3. If the given network is not weakly reversible

Then with any choice of rate constant the reaction cannot have a positive equilibrium steady state.

Deficiency one theorem:à

If a given reaction network satisfy the following:

(i) δθ ≤1, θ=1,2....l δθ:= deficiency of θth linkage class

(ii) 1

l

θ =∑ δθ =δ

And if each linkage class has only one terminal strong linkage class ,

Then, for any (positive) value of rate constant the corresponding CRN has only one steady state for each compatibility class.

S-R graph theoretic approach à S-R graph is a bipartite undirected graph, with two types of nodes, S-node and R-node., it is one-to-one mapping, one S-node correspondence to one species and one R-node corresponding to one reaction. In this approach, reversible reaction treat as a one reaction so it appears only once in the network graph. Different terminology in s-r graph theoretic approach

Complex pair(c-pair):- A pair of edges are called c-pair if this two edge meet at a reaction node and have the same complex label.

o-cycle- A cycle that contains an odd number of c-pair called o-cycle

e-cycle- A cycle that contains en even number of c-pair is called e-cycle. A cycle contains zero c-pair treated as a e-cycle.

Stoichiometric coefficient of edge: Stoichiometric coefficient of species adjacent to the edge in the complex label.

s-cycle- A cycle is called s-cycle if alternatively multiplying and dividing the Stoichiometric coefficient corresponding path of the cycle give final result = 1

S to R intersection – we say two cycles in s-r graph have species to reaction intersection if the common edge of the two cycle constitute a path that begins at a species node and end with a reaction node or they constitute a disjoint union of such path.

Theorem:à

For a given reaction network, corresponding S-R graph fulfils the followings:

(i) Each cycle is an o-cycle or s-cycle (ii) No two e-cycle have an s to r intersection

Then taken with mass action kinetics the network does not have multiple equilibrium point.

In general-

(i) If there are no cycle or all are o cycle, then condition (i) and (ii) are satisfied. (ii) If all Stoichiometric coefficient are one then all cycle are s-cycle ,so condition (i) is

satisfied ,also if no species node is adjacent to three or more reaction node then no two cycle have an s to r intersection so condition (ii) is satisfied.

A1+A

2

2A2

A1+A2

A1+A

2

Example 1: 1 2

2 1 1 2 2 2k k

A A A A← + →

Number of complexes (c) = 3 Number of linkage class (l )=1

Stoichiometric matrix (Г)=1 11 1

− −

Rank of Г (ρ)=1 Deficiency of the network (δ)=c- l –ρ = 3-1-1=1 Differential equation representing this network is following:

jdSdt

=ГR(S) , ∀ Sj∈�

where, R(S)= 1[ 1][ 2]2[ 2][ 2]

k A Ak A A

[ 1] 1[ 1][ 2] 2[ 1][ 2]

[ 2] 1[ 1][ 2] 2[ 1][ 2]

d A k A A k A Adt

d A k A A k A Adt

= −

= − +

From deficiency one theorem we can say that as it has terminal linkage class more than one ({2A1},{2A2}), so it has multiple steady state for each compatibility class. S-R graph theoretic approach:

From this graph it is seen clearly that all the cycle are not either o-cycle or s-cycle.for the cycle

R2-A2-R2 is e-cycle and is not s-cycle.so this network has ability to producing multiple steady state .

A1+A2

A1

A2 R1

R2

A1+A

2

Stoichiometric matrix (Г)=1 11 1

− −

Dimension of the Stoichiometric subspace is one. The table bellow contains steady state value of

initial condition 21

,35

and its compatibility class.

Initial condition

Steady state

Initial condition

Steady state

Initial condition

Steady state

Initial condition

Steady state

21

2.99990.0006

30

30

12

30.0003

03

03

35

7.9990.0005

44

7.99990.0008

26

7.99990.0030

08

08

There is multiple equilibrium point in each compatibility class. So this network can be analysed using both approach.

Example 2:

1 3

24 6

5

......... 1

........ 2

k k

kk k

k

E S ES E P linkageclass

F P FP F S linkageclass

+ → +

+ → +

�

�

Species vector=

ESESPFFP

Stoichiometric matrix:(Γ) =

1 1 1 0 0 01 1 0 0 0 1

1 1 1 0 0 00 0 1 1 1 00 0 0 1 1 10 0 0 1 1 1

− − − − −

− −

Differential equation corresponding reaction is:

jdSdt

=ГR(S) , ∀ Sj∈�

Where R(S)=

1[ ][ ]2[ ]3[ ]4[ ][ ]5[ ]6[ ]

k E Sk ESk ESk F Pk FPk FP

[ ] 1[ ][ ] 2[ ] 3[ ]

[ ] 1[ ][ ] 2[ ] 6[ ]

[ ] 1[ ][ ] 2[ ] 3[ ]

[ ] 3[ ] 4[ ][ ] 5[ ]

[ ] 4[ ][ ] 5[ ] 6[ ]

[ ] 4[ ][ ] 5[ ] 6[ ]

d E k E S k ES k ESdt

d S k E S k ES k FPdt

d ES k E S k ES k ESdt

d P k ES k F P k FPdt

d F k F P k FP k FPdt

d FP k F P k FP k FPdt

= − + +

= − + +

= − −

= − +

= − + +

= − −

For these network no of linkage class(l ) =2

No of complexes (c)= 6

Rank of stoichiometric matrix(ρ)=3

Deficiency=c-l-ρ

=1

Each linkage class has only one terminal linkage class .

Linkage class 1:

No of complex: 4

Stoichiometric metrix:

1 1 11 1 0

1 1 10 0 1

−−

− −

Rank of stoichiometric matrix= 2

So deficiency of linkage class1(δ1)= 4-1-2=1

For linkage class2 no of complexes=4

and stoichiometric matrix is

1 1 11 1 0

1 1 10 0 1

−−

− −

Same as linkage class1,so its deficiency (δ2)is 1

This network is not satisfying deficiency one theorem as δ1+δ2≠ δ

so we cannot say about steady state using deficiency theorem.

S-R graph theoretic analysis:

All Stoichiometric coefficients are one, so all the cycles are s-cycle, condition one is satisfied. R4-S-R1-E-R2-P-R3-F-R4 cycle ,R1-ES-R2-P-R3-FP-R4-S-R1 cycle, R1-ES-R2-E-R1 cycle, R4-FP-R3-F-R4 cycle all are e-cycle , but they have reaction to reaction intersection so condition two also satisfied.

E

P

F

S

ES

FP

R1 R2

R4 R3

E+S

E+S

F+S

E+P

F+S

FP FP

F+P

F+P

E+P

ES ES

Hence applying s-r graph theoretic approach we can say that this network have not multiple equilibrium point. .

Simulation result:

Steady state value for initial condition

345678

and

783210

and their compatibility classes are listed

below(for k1=k2=k3=k4=k5=k6=1).

Initial

condition

Steady

state

Initial

condition

Steady

state

Initial

condition

Steady

state

Initial

condition

Steady

state

:

345678

:

1.5218.5216.4791.5218.5216.479

236678

1.5218.5216.4791.5218.5216.479

444778

1.5218.5216.4791.5218.5216.479

345569

1.5218.5216.4791.5218.5216.479

Initial

condition

Steady

state

Initial

condition

Steady

state

Initial

condition

Steady

state

Initial

condition

Steady

state

:

783210

:

9.1530.1850.84711.120.1520.847

674210

9.1530.1850.84711.120.1520.847

882310

9.1520.1850.84711.120.1520.847

783201

9.1520.1870.85812.100.1410.858

From simulation result it is clear that this network has not multiple equilibrium point for each

compatible class.

2A+B 3A

Example 3:

1

2

2 3k

k

A B A+ �

Total number of complex(c) is 2.

Species vector is AB

Stoichiometric matrix is(Γ)=1 11 1

− −

Differential equation representing this network is following:

jdSdt

=ГR(S) , ∀ Sj∈�

Where R(S)=2

3

1[ ] [ ]2[ ]

k A Bk A

2 3

2 3

[ ] 1[ ] [ ] 2[ ]

[ ] 1[ ] [ ] 2[ ]

d A k A B k Adt

d B k A B k Adt

= −

= − +

Rank of Γ(ρ) is= 1

Here linkage class(l ) = 1

Deficiency is=c-l -ρ

=0

This network is reversible reaction ,so applying zero deficiency we can say that, whatever for any positive value of reaction rate,the system always have one equilibrium point in each compatibility class.

S-R graph analysis:

B A R1 2A+B

in this S-R graph which is related to the above reaction network, has only one cycle which is neither o-cycle nor s-cycle. So applying S-R graph theory approach we can say that this reaction network multiple steady state.

Simulation result:

Steady state value of initial condition22

, and 55

and its compatibility classes are listed

below, when k1=k2=1.

Initial condition

Steady state

Initial condition

Steady state

Initial condition

Steady state

Initial condition

Steady state

22

22

13

22

04

04

40

22

55

55

46

55

37

55

010

010

From result it is clear that there are two steady state values for each compatibility class. So we can say that using zero deficiency theorem we cannot explain this network but using S-R graph theory we can explain its behaviour.