The Kinetic Matrix Method

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Preamble

The Formulation

Example Scheme

Preamble

I guess this method is useful for particularly complex kinetic systems, providing a succint notation and an ease of handling complex sets of kinetic equations.

The Formulation

In the matrix formulation of reaction kinetics, the system of equations that make up a kinetic system are represented by two matrices. We have the state vector, `bb"X"`, a column vector (wikipedia) containing a list of the chemical species in the system, and a rate matrix, `bb"K"`, containing the coefficients as they appear in the kinetic differential equations. The state vector is a vertical list, whereas the rate matrix is square, with a coefficient for each species. We can formulate the kinetic differential equations in terms of these matrices,

`bb"X"' = bb"K"*bb"X"`

The solution to the kinetic matrix follows that of the regular exponential solution, but involves a new matrix, `exp(bb"K"t)`, the state transition matrix,

`bb"X"(t)=exp(bb"K"t)bb"X"(0)`

Which we can express using the Taylor-series expansion,

`exp(bb"K"t)=bb"I"+bb"K"t+(bb"K"t)^2/(2!)+...`

Example I: First-Order Reaction

Let us consider the irreversible first-order process,

`Astackrel(k)rarrB`

Kinetic differential equations:

`A'(t)=-kA`

`B'(t)=kA`

  

`bb"X"= [[A],[B]]`
State vector

  

`bb"K"=[[-k,0],[k,0]]`
Rate matrix

`bb"X"'=bb"K"*bb"X"`
Kinetic matrix

Which has the solution,

`bb"X"=[[A_1],[A_2]]exp(omega t)`

Then we do something to do with eigenvalues,

`bb"X"' = bb"K"*bb"X"=omega bb"X"`

Where `bb"I"` is the unit matrix, `bb"X" = bb"I"*bb"X"`, then

`(bb"K"-omega bb"I")*bb"X"=0`

For there to exist non-trivial solutions, the determinant, `|bb"K"-omega bb"I"|`,  of the matrix must be zero, ie.,

`|bb"K"-omega bb"I"|=0`

and then from somewhere we get the eigenvalues,

`omega_1=0, omega_2=-k`

`"Eigenvalues"(bb"K")=0, -k`

`"Eigenvectors"(bb"K")=[[-k],[0]], [[-1,0],[1,1]]`

Example I: Twice-Consecutive First-Order Reaction

Let us consider the Twice-Consecutive First-Order reaction mechanism,

Scheme:

`Astackrel(k_1)rarrBstackrel(k_2)rarrC`

Kinetic differential equations:

`A'(t)=-k_1A`

`B'(t)=k_1A-k_2B`

`C'(t)=k_2B`

For this mechanism, our state vector is a list, containing three rows.

`bb"X"= [[A],[B],[C]]`
The state vector

Each of the three species has a kinetic differential equation, which may depend on any or all of the concentrations of each species, hence the rate matrix is a 3x3 square matrix. Each row in the rate matrix, `bb"K"`, corresponds to coefficients of the differential equation for species in the corresponding row in the state vector.

`bb"K"= [[-k_1,0,0],[k_1,-k_2,0],[0,k_2,0]]`
The rate matrix

The first row in the matrix corresponds to the species A. Each column in the rate matrix corresponds to a dependence upon a different species, for example, column one is the coefficient of A in the rate equation, A'(t), and this has the figure -k1; column two is the coefficient for the dependance on B, which is zero, since A'(t) is not a function of B; and the third column corresponds to a dependance on C, which is also zero. The second row corresponds to the species B, for which the kinetic differential has a term dependant upon A (with a coefficient of k1), a dependance upon B (with a coefficient of -k2), and no dependance upon C (and so has a figure of 0).

`d/(dt) [[X_1],[X_2],[X_3]]=[[-k_1,0,0],[k_1,-k_2,0],[0,k_2,0]][[X_1],[X_2],[X_3]]`

We can demonstrate that these matrices expand out to give the regular kinetic differential equations. Using the rules of matrix multiplication, for example,

The set of eigenvalues of `bb"K"` are,

`"Eigenvalue" (bb"K")=[[-k_1],[-k_2],[0]]`

ie.,

`lambda_1=-k_1`, `lambda_2=-k_2` and `lambda_3=0`

...and the set of eigenvectors are,

`bb"C"_1^0 = [[k_2-k_1],[k_1],[-k_2]]`, `bb"C"_2^0 = [[0],[1],[-1]]` and `bb"C"_3^0 = [[0],[0],[1]]`

Example I: The unimolecular decay