Module-Based Analysis of Robustness Tradeoffs in the Heat Shock Response System

Using module-based analysis coupled with rigorous mathematical comparisons, we propose that in analogy to control engineering architectures, the complexity of cellular systems and the presence of hierarchical modular structures can be attributed to the necessity of achieving robustness.

What is a modular architecture?

What is protocol?

ExistingModule

New module

New Module

How is a new module added to the existing system?

TCP/IPUSB

What is robustness?

Biological systems maintain their homeostasis against environmental stress, genetic changes and noises.

Time

Par

amet

er Perturbation

What is a tradeoff?

A tradeoff usually refers to losing one quality or aspect of something in return for gaining another quality or aspect.

It implies a decision to be made with full comprehension of both the upside and downside of a particular choice.

(from WIKIPEDIA)

Heat shock response

A universal principle?Robustness tradeoffs generate complexity.

32RNAP DnaK

FtsH

Pfold Punfold

folded mRNA(32)

DnaK gene

Heat Shock

32 gene

FtsH gene

PLANT

ACTUATOR

FF SENSOR

FB SENSORCOMPUTER

heat-activated mRNA(32)

1 Molecular module

2 Functional module

3.Flux module

Hierarchical modular structure

Modular Decomposition in the Heat Shock Response

1. Molecular module RNAP, 32, DnaK FtsH, gene, mRNA,….

2. Functional module PLANT FF SENSOR FB SENSOR COMPUTER ACTUATOR

FF=feedfoward, FB=feedback

32RNAP DnaK

FtsH

Pfold Punfold

folded mRNA(32)

DnaK gene

Heat Shock

32 gene

FtsH gene

heat-activated mRNA(32)

PLANT

FB SENSOR

FF SENSOR

COMPUTER

ACTUATOR

32RNAP DnaK

FtsH

Pfold Punfold

DnaK gene

Heat Shock

FtsH gene

PLANT

ACTUATOR

FB SENSORCOMPUTER

A

folded mRNA(32)

32 gene

FF SENSORheat-activated mRNA

(32)

32RNAP DnaK

FtsH

Pfold Punfold

DnaK gene

Heat Shock

FtsH gene

PLANT

ACTUATOR

FB SENSORCOMPUTER

B

km[1]

folded mRNA(32)

32 gene

FF SENSORheat-activated mRNA

(32)

K[4]

32RNAP DnaK

FtsH

Pfold Punfold

DnaK gene

Heat Shock

FtsH gene

PLANT

ACTUATOR

FB SENSORCOMPUTER

C

K[5]kx[1]

folded mRNA(32)

32 gene

FF SENSORheat-activated mRNA

(32)

K[6]kx[2]

32RNAP DnaK

FtsH

Pfold Punfold

DnaK gene

Heat Shock

FtsH gene

PLANT

ACTUATOR

FB SENSORCOMPUTER

Dfolded mRNA(32)

32 gene

FF SENSORheat-activated mRNA

(32)

3. FLUX Module

FFFeedforward flux module

SEQ-FBSEQ-Feedback flux module

DEG-FBDEG-Feedback flux module

32 amplificationflux module

1. FF: Temperature-induced translation of the rpoH mRNA2. SEQ-FB: DnaK-mediated sequestering 32 3. DEG-FB: FtsH-mediated 32 degradation

32RNAP DnaK

FtsH

Pfold Punfold

folded mRNA(32)

DnaK gene

Heat Shock

32 gene

FtsH gene

PLANT

ACTUATOR

FF SENSOR

FB SENSORCOMPUTER

heat-activated mRNA(32)

32 amplification

SEQ-FB

DEG-FB

FF

Four flux module

Equations

0( ) tt f

dST S F

dt

td f d t

dDK S D

dt

d sf d

f

KdFS F

dt

, ( st

f

F D

)

: ( )foldfold fold

dPK U D K T P

dt

: s f fS D K S D

: u f fU D K U D

: : t fD D U D S D

: t fS S S D

:t fold fP P U U D

Mathematical module decompositionA simple model for the heat shock response

0( ) tt f

d ST S F

d t

td f d t

d DK S D

d t : ( )f o l d

f o l d f o l d

d PK U D K T P

d t

: s f fS D K S D

: t fS S S D

C o m p u t e r F B S e n s o r

A c t u a t o r

F F S e n s o rH e a t S h o c k

: u f fU D K U D

: : t fD D U D S D:t f o l d fP P U U D

P l a n t

+

d sf d

f

Kd FS F

d t

0( ) tt f

d ST S F

d t

td f d t

d DK S D

d t : ( )f o l d

f o l d f o l d

d PK U D K T P

d t

: s f fS D K S D

: t fS S S D

C o m p u t e r F B S e n s o r

A c t u a t o r

F F S e n s o rH e a t S h o c k

: u f fU D K U D

: : t fD D U D S D:t f o l d fP P U U D

P l a n t

+

d sf d

f

Kd FS F

d t

Mathematical functional decomposition of the reduced order heat shock system

0( ) tt f

d ST S F

d t

td f d t

d DK S D

d t

( )st

f

F D

: ( )f o l df o l d f o l d

d PK U D K T P

d t

d sf d

f

Kd FS F

d t

( 1 )

( 2 )

( 3 )

( 4 )0( ) t

t f

d ST S F

d t

td f d t

d DK S D

d t

( )st

f

F D

: ( )f o l df o l d f o l d

d PK U D K T P

d t

d sf d

f

Kd FS F

d t

( 1 )

( 2 )

( 3 )

( 4 )

SEQ-FB

FF

DEG-FB

Mathematical flux decomposition of the reduced order heat shock system

Mathematical system analysis

The main objective of the heat shock response system is to refold denatured proteins upon exposure of higher temperatures by the heat shock proteins (hsps: e.g. chaperone, DnaK, FtsH,…).

1. Response speed

2. Yield for refolded proteins How much proteins are refolded?

3. Efficiency for chaperones How less chaperones are employed for refolding process?

4. Robustness (e.g. sensitivity analysis) Sensitivity of chaperone (DnaK) to parameter uncertainty Resistance of chaperone (DnaK) to noise

Characterization criteria

Virtual knockout mutant

A flux module is removed while conserving the other modules in computers.

A flux module is disabled to explore the function of it.

Mathematical comparison for robustness

Some performances are compared while the others are set to the same.

Wild: SEQ+DEG+FFYield

Efficiency

Response speed

Mutant: SEQ+FF YieldEfficiency

Response speed

==

>

For example, a response speed is compared between wild type and a virtual knockout mutant while the yield and efficiency are set to the same.

SEQ-FB (DnaK-mediated sequestering 32 )

It seems sufficient for refolding proteins. Why other flux modules are added?

1. (FF) Temperature-induced translation of the rpoH mRNA2. (DEG-FB) FtsH-mediated 32 degradation

At least two flux modules are added to the heat shock response.

Time course of 32 and yield

A response time is compared ：　 FF slow SEQ slow SEQ+DEG middle　 SEQ+DEG+FF fast

0

200

400

600

800

1000

0 50 100 150

A

To

tal

32 C

on

cen

tra

tio

n (

nM

)

Time (min)

0.5

0.6

0.7

0.8

0.9

1

1.1

0 50 100 150

C

Yie

ld (

-)Time (min)

0

20

40

60

80

100

0 200 400 600 800 1000

Res

po

nse

Tim

e (m

in)

Total 32 Concentration (nM)

FF

SEQ+FF

SEQ+DEG+FF

FF 　　　　　　 slowSEQ+DEG+FF 　 very fastSEQ 　　 +FF 　 very fast at a high concentration of 32 　　　　

Response time

Robustness of chaperone against parameter uncertainty

Sensitivity analysis

SEQ enhances the robustness (low sensitivity),while neither DEG addition nor FF addition does it.

SEQ+DEG

SEQ+DEG+FF

SEQ0 2 4 6 8 10 12 14 16 18 20 22

02

46

810

1214

0

0.2

0.4

0.6

0.8

1

K[4]

B

km[1]

Se

ns

itiv

ity

0 2 4 6 8 10 12 14 16 18 20 22

02

46

810

1214

0

0.2

0.4

0.6

0.8

1

K[4]

C

km[1]

Se

ns

itiv

ity

0 2 4 6 8 10 12 14 16 18 20 22

02

46

810

1214

0

0.2

0.4

0.6

0.8

1

K[4]

A

km[1]

Se

ns

itiv

ity

Addition of DEG-FB provides the robustness to noise

200

220

240

260

280

300

500 600 700 800 900 1000

Time(min)

To

tal D

naK

Co

nc

entr

ati

on

(nM

)

A

Robustness to noise

0

0.1

0.2

0.3

0.4

0.5

0 200 400 600 800 1000C

V (

-)

Total 32 Concentration (nM)

B

SEQ

SEQ+DEGStochastic simulation

Yield Parameter

Uncertainty

Stochastic fluctuation

Response speed

FF ○ Ｘ Ｘ △

SEQ-FB △ ○ ○ △

DEG-FB

Ｘ △ ○ ○

FF+SEQ-FB +DEG-FB

○ ○ ○ ○

Robustness and Tradeoff

Robustness tradeoffs generate complex regulations.

SEQ

SEQ+DEG

SEQ+DEG+FF

+Fast response+Resistance to noise

SEQ+FF

Two cinarios for the heat shock response evolution

+High yield

Resistance to parameter uncertaintyResistance to noise

+Fast response+High yield

Low 32 concentration in cytoplasm

High 32concnetration in periplasm

32 is very weak.

DnaK Module

LonModule

FtsHModule

Other hsp Modules

32

Interconnected feedback loops

Fragility is generated.

Evolvable architecture of the interconnected feedback

F D Module

B Module

C Module

A Module

F D ModuleC Module

A Module

B Module

Interconnected Feedback Loop

FtsH Gene

FtsH Gene

FtsH

Binding Domain

Promoter for Binding Region for FtsH

Non-Interconnected Feedback (Autogenous Control)

DNA Binding Domain

FtsH

Binding Domain32 32

32 32

38

Protocol for new flux module addition

Hierarchical module architectureRobustness tradeoffs evolve complex systems

Similarity between biology and engineering

Figure Hierarchical Interconnected Feedback Loops

RNAP

54 module

38 module

other factor modules

32

DnaK module

Lonmodule

FtsHmodule

other hsp modules

Expand 32 module

70

Biological systems

Virtual biological systems

Engineering systems

Kurata, 2000

ComparisonIn silicomodeling

Design principle underlying molecular networks (bioalgorithm)In analogy to engineering systems

Strategy for exploring design principles