A biologist and a matrix

The matrix will follow

• Did you know that the 20th century scientist who lay the foundation to the estimation of signals in random noise was sir RA Fisher, a biologist (known for the Fisher information matrix?)

• This lecture is for biologists. It is meant to be non rigorous and intuitive

• If you are not from life sciences please refrain from correcting mathematical inaccuracies. Kill me in the break

Agenda• Many times we encounter highly complex experimental data (e.g.

cell signaling, neuron firing patterns) or want to model a complex process.

• Matrices can be used to represent functions on these data records, or to model the process creating the data.

• 3 examples in this lecture:– From molecular biology- cell signaling. how do we use matrix algebra to

find the more interesting pathway? – From biochemistry- ligand / receptor dynamics. How do we use algebra

to simulate it?– From Evolutional biology- how do we identify evolutional constraints?

• All the math is in the presentation so you don’t need to copy

A “function” has:

•A source

•A target

•A rule to go from the source to the target

)(xfy 0

0.5

1

1.5

2

2.5

-0.5 0 0.5 1 1.5

3.02 xy

The source and target can be 2 or 3 dimensional

),( yxfH H can be a topographic map, for example

For each point on the map we assign a number

)(),,( tfZYX The other way around:

The orbit describes the movement of the planets as a function of time in 3-D

Lets consider functions from 2-D to 2-D:

),(

),(

),(),(

yxfY

yxfX

YXyxf

y

x

y

xIf we can write:

fyexdY

cybxaX

Then f is called linear

2 important functions from 2-D to 2-D

y

x

The function that takes every point to (0,0) : the zero function

The function that doesn’t do anything : the unity function

010

001

yxY

yxX

000

000

yxY

yxX

zero matrix

yxY

yxX

00

00

0

0

00

00

y

x

Y

X

yxY

yxX

10

01

y

x

y

x

Y

X

10

01

The matrix notion:

unity matrix

•Every linear function 2-D to 2-D can be written by a 2x2 matrix

•Every 2x2 matrix represent a linear function from 2-D to 2-D

cossin

sincos

yxY

yxX

y

x

Y

X

cossin

sincos

Another example: a rotation matrix

More examples: reflection, compression, stretching…

y

x

y

x

Y

X

cossin

sincos

Ф

In a rotation, the vector’s length remain the same

0

0

00

00

y

x

Y

X00

vV

The matrix notion:

y

x

y

x

Y

X

10

01vvIdV

???/ 1 vvvvId

10

011

An inverted matrix:

Y

X

y

x

y

x

Y

X1

10

01

10

01

?1

Y

X

y

x

y

x

Y

X

Matrix math: only square matrices can be inverted, and not even all of them

zero matrix inverse?

?00

00

00

001

Y

X

y

x

y

x

Y

X

unity matrix inverse?

A vector which is only scaled by a specific matrix operation is called an eigenvector. The scaling factor is called an eigenvalue .

y

x

vvA

Anyway, one thing remains: the reversibility of a matrix depends on its eigenvalues. Invertible matrix no zero eigenvalues, λ≠0.

There is an “intuitive” explanation: inversion, reverse, means that you can go back from Y to X.

The vector (0,0) always stay (0,0) after matrix operation.

if another vector, other than (0,0) is transformed to (0,0) by the matrix, we cant go back from (0,0) to the vector that produced it- we just cant tell which one it was. If there is an eigenvalue 0 then infinite number of vectors can be the source of (0,0).

What is the physical meaning of the eigenvectors/ values?

For every use of matrices there is a different meaning. We will see an example.

A major task of engineering:

make the data easy on the eyes[1]

• Biology example: cell signaling.

• Many signals, many observations = big matrix, big mess

• Transform this matrix into something we can look at, by choosing the best x and y axes

[1] Kevin A. Janes and Michael B. Yaffe, Data-driven modeling of signal-transduction networks, Nature Reviews Molecular Cell Biology 7, 820-828 (November 2006)

“The paradox for systems biology is that these large data sets by themselves often bring more confusion than understanding” [1]

The idea: arrange the rows and columns of the matrix in a way that reveals biological meaning

The example: measure the co-variance (how 2 cell signals change “together”), to create a matrix:

•For the eigenvectors, the matrix just change the vector size (multiply by the eigenvalue)•The meaning- the eigenvector identifies a fraction of each measured signal that varies with all the others [1]•The eigenvalue quantifies the strength of this global co-variation

•Biggest eigenvalues of C correspond to the most informative collection of signals- the ones that behave “together”

•Choose for example only the biggest 2, and use them as the X and Y axis

•How do we change the existing data vectors to the new axes?

•We project!y

x

Ф

r

cos

sin

ry

rx

BTW, this method is called Principal Components analysis (PCA)

Another use of matrices: advance in time

0

1

1

0

01

10)2(

1

0

0

1

01

10)(

0

1

01

10,

tv

tv

v

AxYyX

)()( tvAttv

exampley

x

Use of matrices: propagator function- advance in time

1

1

1

1

01

10)(

1

1

1

1

01

10)(

tv

tv

y

x45○

What is the eigenvalues of the eigenvectors?

The 2 eigenvectors can be thought of 2 modes of movement in the space- one motionless, the other ‘jumps’ 180 degrees.

And if we build a new vector, a combination of the 2 eigenvectors?

Combination of eigenvector with non-eigenvector

1

2

2

1

01

10)2(

2

1

1

2

01

10)(

1

2

0

1

1

1

tv

tv

v

y

x

What will happen if ?

0

2

12

10

A

Summary:

When the matrix is a propagator the eigenvectors with eigenvalue 1 are the stable states (along side 0)

When the eigenvalues are less than one the system will decay to 0

When the eigenvalues are higher than one the system will grow and grow…

What if we want to check the system state after many time steps?

n

tnv

tv

3.07.0

7.03.0)(

3.07.0

7.03.0

3.07.0

7.03.0

3.07.0

7.03.0)2(

2

How do we calculate the matrix power?

Using the eigenvectors, we can write the matrix as a multiplication of 3 matrices:

5.05.0

5.05.0

4.00

01

5.05.0

5.05.0

3.07.0

7.03.0

10

01

5.05.0

5.05.0

5.05.0

5.05.0

5.05.0

5.05.0

4.00

01

5.05.0

5.05.0

3.07.0

7.03.0

5.0

5.02,

5.0

5.01

n

nn

vv

What can such matrix mean?

- Ligand / receptor binding state, and next state probabilities[2]

Capture state

Free state

0.7

0.3

0.7

0.3

)0,(

)0,(

7.03.0

3.07.0

),(

),(

freeP

captureP

tfreeP

tcaptureP

[2], A. Hassibi, S. Zahedi, R. Navid, R. W. Dutton, and T. H. Lee, Biological Shot-noise and Quantum-Limited SNR in Affinity-Based Biosensor, Journal of Applied Physics, 97-1, (2005).

?1

0

7.03.0

3.07.0

),(

),(

58.0

42.0

1

0

7.03.0

3.07.0

7.03.0

3.07.0

)2,(

)2,(

7.0

3.0

1

0

7.03.0

3.07.0

),(

),(

freeP

captureP

tfreeP

tcaptureP

tfreeP

tcaptureP

For example, assume all ligands are free at time zero:

As only the eigenvector of 1 survives (0.4 mode goes down to zero), we will be left with a uniform probability of (½, ½)- half of the ligand molecules are captured and half are free at steady state

Another example: Evolutionary Biology and genetics“evolutionary biology rests firmly on a foundation of linear algebra”[3]

•Observations are made on the covariance matrix of traits denoted G

•A genetic constraint is a factor that effects the direction of evolution or prevents adaptation

•Genetic correlation that show no variance in a direction of selection will constrain the evolution in that direction. How can we see it in the matrix?

[3], M. W. Blows, A tale of two matrices: multivariate approaches in evolutionary biology, Journal of Evolutionary

Biology , Volume 20 Issue 1 Page 1-8, (January 2007)

A zero eigenvalue