L 7: Linear Systems and Metabolic Networks

Linear Equations

• Form

• System1 1 2 2 n na x a x a x b

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

n n

n n

m m mn n m

a x a x a x b

a x a x a x b

a x a x a x b

Linear Systems

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

n n

n n

m m mn n m

a x a x a x b

a x a x a x b

a x a x a x b

Ax b

11 12 1 1

21 22 2 2

1 2

n

n

m m mn m

a a a b

a a a b

a a a b

A b

Vocabulary

• If b=0, then system is homogeneous

• If a solution (values of x that satisfy equation) exists, then system is consistent, else it is inconsistent.

Solve system using Gaussian Elimination

• Form Augmented Matrix, • Row equivalence, can scale rows and add and

subtract multiples to transform matrix

11 12 1 1 1

21 22 2 2 2

1 2

1 0 0

0 1 0

0 0 1

n

n

m m mn m n

a a a b d

a a a b dm n

a a a b d

11 12 1 1 1

21 22 2 2 2

1 2

1 0 0

0 1 0

0 1 1

n

n

m m mn m m

a a a b d

a a a b dm n

a a a b d

1

211 12 1 1

21 22 2 2

1 2

1 0 0

0 1

0

0 0 1

0 0 0 0

0 0 0 0

n

nn

m m mn m

d

da a a b

a a a bm nd

a a a b

Underdetermined: more unknowns, than equations, multiple solutions

Overdetermined: fewer unknowns, than equations, if rows all independent, then no solution

Linear Dependency

• Vectors are linearly independent iff

• has the trivial solution that all the coefficients are equal to zero

• If m>n, then vectors are dependent

1 1

1

0

, , x1m m

m

a a

vectors are size n

x x

x x

Subspace of a vector space

• Defn: Subspace S of Vn

– Zero vector belongs to S– If two vectors belong to S, then their sum belongs to S– If one vector belongs to S then its scale multiple

belongs to S

• Defn: Basis of S: if a set of vectors are linearly independent and they can represent every vector in the subspace, then they form a basis of S

• The number of vectors making up a basis is called the dimension of S, dim S < n

Rank

• Rank of a matrix is the number of linearly independent columns or rows in A of size mxn. Rank A < min(m,n).

Inverse Matrix

• Cannot divide by a matrix

• For square matrices, can find inverse

• If no inverse exists, A is called singular.

• Other useful facts:

1 1n

AA I A A

11

1 1 1

n

A I

AB B A

Eigenvectors and Eigenvalues

• Definition, let A be an nxn square matrix. If is a complex number and b is a non-zero complex vector (nx1) satisfying: Ab=b

• Then b is called an eigenvector of A and is called an eigenvalue.

• Can solve by finding roots of the characteristic equation (3.2.1.5)

0nDet A I

Linear ODEs

1 11 1 12 2 1 1

2 21 1 22 2 2 2

1 1 2 2

n n

n n

n n n n n n

x a x a x a x z

x a x a x a x z

x a x a x a x z

x Ax z

Steady-state Solution

• Under steady state conditions

• Need to find x:

0 x Ax z

Ax z

1x A z

Time Course

• Take a first order, linear, homogeneous ODE:

• Solution is an exponential of the form

• Put into equation, solve for constant using ICs gives:

111 1

dxa x

dt

1 1tx t b e

1101 1

a tx t x e

Effect of exponential power

• What happens for different values of a11?

• Options: if system is perturbed– Stable- system goes to a steady-state– Unstable: system leaves steady-state– Metastable: system is indifferent

1101 1

a tx t x e

Matrix Time Course

• Take a first order, linear, homogeneous ODE:

• Solution is an exponential of the form

• General solution:

x Ax

tt ex b

1

i

nt

i ii

t c e

x b

Why are linear systems so important?

• Can solve it, analytically and via computer

• Gaussian Elimination at steady state

• Properties are well-known

• BUT: world is nonlinear, see systems of equations from simple systems that we have already looked at

Linearization

• Autonomous Systems: does not explicitly depend on time (dx/dt=f(x,p))

• Approximate the change in system close to a set point or steady state with a linear equation.

• It is good in a range around that point, not everywhere

Linearization• At steady state, look at deviation:

• Use Taylor’s Expansion to approximate:

de tde t e t

dt dt x f x x

1 1

2

11 1 1

, .

1, .

2

ii n n

n n ni i

i n j j kj j kj j k

de tf x e x e

dt

f ff x x e e e HOT

x x x

Linearization• First term is zero by SS assumption,

assume H.O.T.s are small, so left with first order terms

1 1

n ni i

j ij jj jj

de t fe a e

dt x

Stoichiometric Matrices

• Look at substances that are conserved in system, mass and flow

• Coefficients are proportion of substrate and product

1 2 2S S P

1 1 2

1 1 2

1 2, , 2 ,dS dS dP

v v vdt dt dt

Stoichiometric Network

• Matrix with m substrates and r reactions

• N ={nij} is the matrix of size mxr

1

1,r

i ij jj

S n v i m

S = Nv

External fluxesConventions are left to right and top down.

Sv 0

Column/Row Operations

• System can be thought of as operating in row or column space

ri

ci

ith row of

ith column of

a A

a A

Subspaces of Linear Systems