Modelling of mass action kinetics Modelling of mass action kinetics Outline

Outline of Topics

Modelling of mass action kinetic Introduction to mass action kinetics Some concepts From reaction networks to ODEs Shan He Example1: A simple reaction network School for Computational Science University of Birmingham Example2: Enzyme catalysis reactions Module 06-23836: Computational Modelling with MATLAB Assignment

Modelling of mass action kinetics Modelling of mass action kinetics Introduction to mass action kinetics Introduction to mass action kinetics Some concepts Some concepts Mass action kinetics Chemical Reaction network

◮ A familiar example:

◮ k Definition : ” Mass action describes the behavior of reactants 2H2 + O2 −→ 2H2O. (1) and products in an elementary chemical reaction. Mass action ◮ reactants kinetics describes this behavior as an equation where the The quantities on the left hand side are the . The products k velocity or rate of a chemical reaction is directly proportional ones on the right hand side are the . denotes the reaction rate . to the concentration of the reactants. ” ◮ The equation above means after some collisions of 2 H and a ◮ Applications : Used to describe the dynamics of chemical 2 O , a transition state is formed which rapidly leads to the reaction networks in biochemical and cellular systems. 2 product 2 H2O. ◮ Some examples: metabolic flux analysis, cell signalling ◮ The reactants and products are collectively called species of network, etc. the reaction. ◮ Chemical reaction networks : A chemical reaction network is a finite set of reactions among a finite set of chemical species.

Modelling of mass action kinetics Modelling of mass action kinetics Introduction to mass action kinetics Introduction to mass action kinetics From reaction networks to ODEs From reaction networks to ODEs Transform chemical reactions Transform chemical reactions

◮ The reaction equation (1) can be re-written as ◮ The equation (2) can be compacted into matrix-vector 3 3 k notation: Aj Xj −→ Bj Xj (2) Xj=1 Xj=1 AX −→k BX where X1, X2 and X3 denote the species H2, O2 and H2O, where A = [ A1 A2 A3] = [2 1 0], X = [ X1 X2 X3], and respectively; A1 = 2, A2 = 1, A3 = 0, B1 = 0, B2 = 0 and B = [ B1 B2 B3] = [0 0 2] B3 = 2 are the stoichiometric coefficients ; and k denotes the reaction rate . Modelling of mass action kinetics Modelling of mass action kinetics Introduction to mass action kinetics Introduction to mass action kinetics From reaction networks to ODEs From reaction networks to ODEs Reversible reaction Reversible reaction

◮ Consider the reversible reaction:

k ◮ 1 Let X1, X2, X3 and X4 denote the species Na 2CO 3, CaCl 2, Na 2CO 3 + CaCl 2 ⇋ CaCO 3 + 2 NaCl (3) k2 CaCO 3 and 2 NaCl , respectively, then the equation (3) can be ◮ Can be written as the forward and backward reactions: written as:

k k1 X X −→ 1 X X , Na 2CO 3 + CaCl 2 −→ CaCO 3 + 2 NaCl 1 + 2 3 + 2 4

k k2 X X −→ 2 X X , CaCO 3 + 2 NaCl −→ Na 2CO 3 + CaCl 2 3 + 2 4 1 + 2

where k1 and k2 are the reaction rates for the forward and backward reactions, respectively.

Modelling of mass action kinetics Modelling of mass action kinetics Introduction to mass action kinetics Introduction to mass action kinetics From reaction networks to ODEs From reaction networks to ODEs Reversible reaction General form of reaction network

◮ ◮ T Let’s consider multiple chemical reactions of s species Let X = [ X1 X2 X3 X4] and k = [ k1 k2], and X1, ..., Xs , where s ≥ 1, whose iterations are governed by r 1 1 0 0 0 0 1 2 reactions, where r ≥ 1, the general form of reaction network: A = , B = 0 0 1 2  1 1 0 0  s s ki Aij Xj −→ Bij Xj , i = 1 ,..., r, (4) ◮ Each element in A and B, e.g., Aij and Bij is called Xj=1 Xj=1 stoichiometric coefficient. ◮ where for i = 1 ,..., r,, ki > 0 is the reaction rate of the ith Aij and Bij are non-negative integers. s s reaction, and j Aij and j Bij are the reactant and ◮ =1 =1 Formulate the kinetic equations: product of the Pith reaction,P respectively. 4 4 ◮ General matrix-vector form of reaction network: ki Aij Xj −→ Bij Xj , i = 1 , 2 k Xj=1 Xj=1 AX −→ BX (5)

where A = [ Aij ] and A = [ Aij ] are r × s non-negative matrices.

Modelling of mass action kinetics Modelling of mass action kinetics Introduction to mass action kinetics Introduction to mass action kinetics From reaction networks to ODEs From reaction networks to ODEs The law of mass action Vector-matrix exponentiation

◮ Law of mass action : “the rate of a chemical reaction is proportional to the product of the concentrations of the ◮ Vector-matrix exponentiation: for a vector T Rs Rr×s reactants” . x(t) = [ x1,..., xs ] ∈ and nongetative A = [ Aij ] ∈ , xA denotes the element of Rr whose ith component for ◮ For general matrix-vector form of reaction network: Ai1 Ai2 Ais i = 1 ,..., r is the product x1 x2 . . . xs . k AX −→ BX (6) ◮ For example: x 1 2 ◮ Let xj (t) denote the concentration of species Xj at time t and x = 1 , A = , T x 3 4 x(t) = [ x1(t),..., xs (t)] , by the law of mass action:  2   then dx(t) = ( B − A)T [k ◦ xA(t)] (7) dt 2 A x1x2 x = 3 4 . where the notation ◦ denotes component-wise multiplication x1 x2  and the notation xA(t) denotes vector-matrix exponentiation. Modelling of mass action kinetics Modelling of mass action kinetics Introduction to mass action kinetics Example1: A simple reaction network From reaction networks to ODEs Stoichiometric matrix Example 1 ◮ We usually represent complex reactions in a more compact ◮ Consider the reaction network: form called the stoichiometry matrix N: k1 X1 −→ X2 (10) N = ( B − A)T ◮ If a reaction network has r reactions and s species then the k2 X −→ X (11) stoichiometry matrix N ∈ Rs×r . 2 1 ◮ The kinetic equation of a reaction network: ◮ s = 2, r = 2 and write down A and B

dx(t) 1 0 0 1 = N[k ◦ xA(t)] (8) A = , B = dt 0 1  1 0  ◮ Defining react rate matrix: K , diag (k1,..., kr ), equation (8) ◮ Calculate the stoichiometry matrix N: can be written as: T −1 1 dx(t) A N = ( B − A) = = NKx (t) (9) 1 −1 dt  

Modelling of mass action kinetics Modelling of mass action kinetics Example1: A simple reaction network Example1: A simple reaction network

Example 1 Example 1 ◮ From equation (9), the kinetic equation of the reaction network: dx(t) = NKx A(t) ◮ Given: dt x1(t) 1 0 x(t) = , A = −1 1 k1 0 x1(t) x2(t) 0 1  =  1 −1  0 k2  x2(t) ◮ Calculate Vector-matrix exponentiation: −k k x (t) = 1 2 1 (12)  k1 −k2  x2(t) A x1(t) x (t) = ◮ x2(t) We can also write the kinetic equations as: dx (t) 1 = −k x (t) + k x (t) dt 1 1 2 2 dx (t) 2 = k x (t) − k x (t) dt 1 1 2 2

Modelling of mass action kinetics Modelling of mass action kinetics Example2: Enzyme catalysis reactions Example2: Enzyme catalysis reactions

Example 2: Enzyme catalysis reactions Example 2: Enzyme catalysis reactions

◮ ◮ Reaction of a substrate S and an enzyme E to produce a Write down A and B: product P by means of an intermediate species C: 1 0 1 0

k1 k A = 0 1 0 0  S + E ⇋ C −→ 3 P + E k 0 1 0 0 2 ,   ◮ s = 4 and r = 3. Let X1 = S, X2 = C, X3 = S, X4 = P, the 0 1 0 0 reaction network can be written as: B = 1 0 1 0  0 0 1 1 k X + X −→ 1 X   1 3 2 ◮ Calculate the stoichiometry matrix N:

k2 −1 1 0 X2 −→ X1 + X3 T  1 −1 −1 N = ( B − A) = −1 1 1 k3   X2 −→ X3 + X4  0 0 1    Modelling of mass action kinetics Modelling of mass action kinetics Example2: Enzyme catalysis reactions Example2: Enzyme catalysis reactions

Example 2: Enzyme catalysis reactions Example 2: Enzyme catalysis reactions

◮ Calculate Vector-matrix exponentiation: xA(t):

x1(t)x3(t) 1 function dx = EnzymeCatalysis(t,x) A x (t) =  x2(t)  2 x (t) 3 k1 = 1; k2 = 1; k3 = 1.5;  2  4 ◮ From equation (9), the kinetic equation of the reaction 5 N = [-1 1 0; 1 -1 -1; -1 1 1; 0 0 1]; network: 6 K = diag([k1 k2 k3]); dx(t) 7 X = [x(1) *x(3); x(2); x(2)] = NKx A(t) dt 8 −1 1 0 9 dx = N *K*X; k1x1(t)x3(t)  1 −1 −1 10 end =  k2x2(t)  (13) −1 1 1   k x (t)  0 0 1   3 2   

Modelling of mass action kinetics Modelling of mass action kinetics Example2: Enzyme catalysis reactions Example2: Enzyme catalysis reactions

Example 2: Enzyme catalysis reactions Example 2: Enzyme catalysis reactions

◮ The kinetic equation can also be written as: dx 1 function dx = EnzymeCatalysis(t,x) 1 = k x − k x x dt 2 2 1 1 3 2 3 k1 = 1; k2 = 1; k3 = 1.5; dx 2 4 = −(k2 + k3)x2 + k1x1x3 dt 5 dx = [k2 *x(2) - k1 *x(1)*x(3); 6 -(k2+k3)*x(2) + k1 *x(1)*x(3); dx 3 = ( k + k )x − k x (t)x 7 (k2 + k3) *x(2) - k1 *x(1)*x(3); dt 1 3 2 1 1 3 8 k3 *x(2)] dx 9 end 4 = k x dt 3 2

Modelling of mass action kinetics Modelling of mass action kinetics Example2: Enzyme catalysis reactions Assignment

Receipt for modelling of mass action kinetics Assignment

Implement the following so-called minimal oscillating mass action kinetics system: Steps: ◮ k1 Step 1: Write down the reactant/product stoichiometric X1 −→ 2X1 (14) coefficients matrices: A, B. T ◮ Step 2: Calculate the stoichiometric matrix: N = ( B − A) . k2 X1 + X2 −→ 2X2 (15) ◮ Step 3: Calculate the Vector-matrix exponentiation: xA(t). dx(t) A ◮ Step 4: Put them together: = NKx (t), where k3 dt X2 −→ 0 (16) K , diag (k1,..., kr ). Question: Write down the corresponding stoichiometric matrix and kinetic equations. Implement the MATLAB ODE function and run it. Can you suggest what is this system?