Protein Networks

Week 5

Linear Response• A simple example of protein

dynamics: protein synthesis and degradation

• Using the law of mass action, we can write the rate equation.

• S = signal strength (e.g. concentration of mRNA)

• R = response magnitude (e.g. concentration of protein)

Linear Response

Protein Cycles20% of the human protein-coding genes encode components of signalingpathways, including transmembrane proteins, guanine-nucleotide binding proteins (G proteins), kinases, phosphatases and proteases.

The identification of 518 putative protein kinase genes and 130 proteinphosphatases suggests that reversible protein phosphorylation is a central regulatory element of most cellular functions.

Abundance of Kinases

Species # of putative kinases

Saccharomyces cerevisiae 121

Drosophila melanogaster 319

C. elegans 437

Arabidopsis thal 1049

Human 518

Data from http://www.kinexus.ca

The Simple Cascade

v 1

v2

Conservation laws

Hyperbolic Response

Assume linear kinetics

Hyperbolic Response

Sigmoidal Response

Assume saturable kinetics

Sigmoidal Response

Assume saturable kinetics

Sigmoidal ResponseMemoryless Switch

Assume saturable kinetics

Fundamental Properties

X

E1

E2

Ultrasensitivity

Kms = 0.5

Fundamental Properties

X

E1

E2

Ultrasensitivity

Kms = 0.1

Fundamental Properties

X

E1

E2

Ultrasensitivity

Kms = 0.02

Colle

ctor

Cur

rent

Base Current

Input

Input

Out

put

Device Analogs

Digital CircuitsIn ultrasensitive mode, cascades can be

used to build Boolean circuits.

Basic Logic GatesNAND Gate – fundamental building block of all logic circuits

A B C

0 0 1

0 1 1

1 0 1

1 1 0

CBA

Basic Logic GatesNOT Gate

A B

0 1

1 0BA

BA

Basic Logic GatesNOT Gate

A

B

Ring Oscillator

NAND Gate

CBA

B

C

A

Memory UnitsBasic flip-flop

R = resetS = setQ = output

Memory Units

Clocked RS flip-flop

R = resetS = setC = clockQ = output

Counters

0 0 0 0

1 0 0 0

0 1 0 0

1 1 0 0

0 0 1 0

1 0 1 0

0 1 1 0

1 1 1 0

Binary Counter

etc

Clock RS flip-flop

Clock input

Arithmetic

Half Adder (No carry input)A B Sum Carry

0 0 0 0

0 1 1 0

1 0 1 0

1 1 0 1

Sigmoidal ResponseMultiple Cycles

Assume linear kinetics

S3

Sigmoidal ResponseBistable Switches

Cell-signalling dynamics in time and spaceBoris N. Kholodenko Nature Reviews Molecular Cell Biology 7, 165-176 (March 2006) |

Sigmoidal ResponseOscillators

Cell-signalling dynamics in time and spaceBoris N. Kholodenko Nature Reviews Molecular Cell Biology 7, 165-176 (March 2006) |

Sigmoidal ResponseOscillators

Cell-signalling dynamics in time and spaceBoris N. Kholodenko Nature Reviews Molecular Cell Biology 7, 165-176 (March 2006) |

Sigmoidal ResponseOscillators

Cell-signalling dynamics in time and space Boris N. Kholodenko Nature Reviews Molecular Cell Biology 7, 165-176 (March 2006) |

Amplifiers – basic amplifier

Ktesibios, 270BC invented the float regulator to maintain a constant water flow which was in turn used as a measure of time.

http://www.control-systems.net/recursos/mapa.htm

Amplifiers – basic amplifier

Centrifugal fly-ball governor, introduced by Watt in 1788 to control the speed of the new steam engines.

http://visite.artsetmetiers.free.fr/watt.html

By 1868 it is estimated that 75,000 governors were in operation in England

Amplifiers – basic amplifier

Harold Black in 1927, invented the first feedback amplifier in order to solve the problem of signal distortion when American Telephone and Telegraph wanted to lay telephone lines all the way from the east to the west coast.

Amplifiers – basic amplifier

Amplifier (A)

Feedback (k)

Output (y)Input (u)

e = error

-

+ e

Amplifiers – basic amplifier

Amplifier (A)

Feedback (k)

Output (y)Input (u)e

-

+

Amplifiers – basic amplifier

Amplifier (A)

Feedback (k)

Output (y)Input (u)e

-

+

Amplifiers – basic amplifier

Amplifier (A)

Feedback (k)

Output (y)Input (u)e

-

+

Amplifiers – basic amplifier

Amplifier (A)

Feedback (k)

Output (y)Input (u)e

If kA > 0 then

-

+

Amplifiers – basic amplifier

Amplifier (A)

Feedback (k)

Output (y)Input (u)

1. Robust to variation in amplifier characteristics2. Linearization of the amplifier response3. Amplification of signal4. Preferential changes in input and output impedances5. Improved frequency response

741 op amp

Feedback – basic amplifier

Amplifiers – basic amplifier

Provided the feedback is below thethreshold to cause oscillations, feedbacksystems can behave as robust amplifiers.

Amplifiers – Synthetic Amplifier

Cascades as Noise Filters

Cascades can act assignal noise filters in the most sensitive region

Output

-50

-40

-30

-20

-10

0

10

20

30

0.001 0.01 0.1 1 10 100

Frequency

Mag

nitu

de

V1 = 1.7

V1 = 3.06

V1 = 3.4

V1 = 3.74

V1 = 5.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2

V1 / V2

P2 /

(P1

+ P2

)

Why? Frequency Analysis

-35

-30

-25

-20

-15

-10

-5

00 0.5 1 1.5 2

V1/V2

Nr(

dv/d

s)L

Jacobian

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2

V1 / V2

P2 /

(P1

+ P2

)

Homeostatic Systems – perfect adaptation

Simultaneous stimulation of input and output steps

ThursdaySimulating other kinds of ‘computational’ behavior

1. Adaptive systems2. Amplifiers and feedback regulation3. Feed-forward networks4. Low an high pass filter