Simplicity is Complexity in Simplicity is Complexity in MasqueradeMasquerade

Michael A. SavageauThe University of California, Davis

July 2004

Complexity is Not Complexity is Not Simplicity in Simplicity in

MasqueradeMasquerade

-- E. Yates

Simplicity is Simplicity is Complexity in Complexity in

MasqueradeMasquerade

One of JohnOne of John’’s Principles of s Principles of Good DesignGood Design

OutsideTransparent to the userWorks like magicSimple

InsideInscrutable to the userRich in robust control structuresComplex

A Comparative Nonlinear Approach to the A Comparative Nonlinear Approach to the Elucidation of Function, Design and Elucidation of Function, Design and Evolution of Biochemical SystemsEvolution of Biochemical Systems

MotivationImportance of comparisonWell-controlled comparisons

MethodologyModeling strategiesCanonical nonlinear formalismMathematically controlled comparisons

Biological design principlesAnticipatory control in biosynthetic pathwaysCoupling of elementary gene circuitsDemand for alternative modes of gene control

OutlineOutline

Importance of ComparisonImportance of ComparisonWhy is there something and not nothing?Why is there something and not something else?Comparison is central to biology

Experimental investigationEvolutionOptimization

( foundations)

Single changes in isogenic background

Single changes have multiple consequences

Secondary consequences can mask primary consequencesState changes in nonlinear systems can have dramatic consequences

Difficulties in Making Difficulties in Making WellWell--Controlled ComparisonsControlled Comparisons

( information)

( structure)

( design)

(complexity)

ExampleExample

Substrate Intermediate Product

NA mRNA

AA Enzymes

Mutant lacking inhibition

vP = VP 1+[ product]KP

⎛

⎝ ⎜

⎞

⎠ ⎟

2⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

−1

≈ VP / 2 Stable vP =VP Unstable

vP =VP / 2 StablevP = VP 1+[ product]KP

⎛

⎝ ⎜

⎞

⎠ ⎟

2⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

−1

≈ VP / 2 Stable

End-product inhibition

Substrate Intermediate Product

NA mRNA

AA Enzymes

MotivationImportance of comparisonWell-controlled comparisons

MethodologyModeling strategiesCanonical nonlinear formalismMathematically controlled comparisons

Biological design principlesAnticipatory control in biosynthetic pathwaysCoupling of elementary gene circuitsDemand for alternative modes of gene control

OutlineOutline

Two Modeling StrategiesTwo Modeling StrategiesSpecific system

Identify a specific system of interestAssemble available information and formulate a modelEstimate parameter values and simulate known behaviorsSuccessful outcome

o Mimic real systemo Predict additional behaviors

Class of systemsIdentify class with many membersAbstract essential characteristics and formulate a modelSymbolic analysis and statistical samplingSuccessful outcome

o Understand the basis for nearly universal designso Discover rules for distinguishing alternative designs

Interlaced Levels of Description for Interlaced Levels of Description for a Chemical Reactiona Chemical Reaction

QM wave function

Potential energy function

Probability distribution function

Rate law function

Boolean function

Discrete/Stochastic

Continuous/Deterministic

Discrete/Stochastic

Continuous/Deterministic

Discrete/Deterministic

Tim

e/N

umbe

r Sca

le

Smal

lLa

rge

( scale, physics)

PowerPower--Law FormalismLaw FormalismdXidt

= α ikk =1

r

∑ Xjgijk

j=1

n

∏ − β ikk=1

r

∑ Xjhijk

j=1

n

∏

Canonical from Four Different Perspectives FundamentalRecastLocalPiece-wise

M. Savageau, Chaos 11: 142 (2001)

( foundations)( scale)

( physics)

( design)

Methodology Implications of the Methodology Implications of the Canonical PowerCanonical Power--Law FormalismLaw Formalism

Fundamental representationReference for detailed kinetic descriptionsGeneralization of mass-action kinetics

Local representationRegular mathematical structureReasonable degree of local accuracy

Piece-wise representationRegular mathematical structureReasonable degree of global accuracy

Recast representationGlobally equivalent Converts implicit equations into explicit equationsEfficient solver for ODEs and algebraic equations

Irvine & Savageau, SIAM J. Numerical Anal. 27:704 (1990)Mueller, Burns & Savageau, Appl. Math. Comput. 90:167 (1998)

dx / dt = 0.343 − (y+ 17.15)e− x x(0) = 3.85

dy / dt = e− x − (50 + z) y(0) = 7.16dz / dt = 1.82 + (y − 9.75)z z(0) = 7.98

Recast RepresentationRecast Representation

dx1 / dt = 0.343x1 − x2 x1(0) = 46.87dx2 / dt = x1 − x3x4 x2(0) = 24.31

dx3 / dt = 1346.82x4−1 − 50x2x4

−1 x3(0) = 57.98dx4 / dt = x2x4 − 26.9x4 x4 (0) = 1

Savageau & Voit, Math. Biosci. 87:83 (1987)

X1=eX, X2=y+17.15, and X3X4=z+50

Global Accuracy of Recast RepresentationsGlobal Accuracy of Recast Representations

0

15

30

45

60

75

-50 0

X1

50 100 150 200 250

Time

X2

35

40

45

50

55

60

65

70

75

0 5 10 15 20 25 30

X2

35 40

X1

Mathematically Controlled ComparisonMathematically Controlled ComparisonTwo designs are represented in a canonical nonlinear formalismDifferences are restricted to a single specific processOne design is chosen as the referenceInternal equivalence is maintainedExternal equivalence is imposedThe systems are characterized by rigorous mathematical and computer analysisComparisons are made on the basis of quantitative criteria for functional effectiveness

( foundations)

MotivationImportance of comparisonWell-controlled comparisons

MethodologyModeling strategiesCanonical nonlinear formalismMathematically controlled comparisons

Biological design principlesAnticipatory control in biosynthetic pathwaysCoupling of elementary gene circuitsDemand for alternative modes of gene control

OutlineOutline

Simple Systemic Behavior Simple Systemic Behavior ----Autocatalytic Growth of BacteriaAutocatalytic Growth of Bacteria

Cells

•

•

••••

•• • •

•

ln O

D

Time

SteadySteady--State GrowthState Growth

Numerous models

ln O

D

TimeCit

Addition of a NutrientAddition of a Nutrient

••

••••

•• • •

•

None predict this behavior!

Elucidation of the Underlying Elucidation of the Underlying Mechanisms Reveals Layers of Mechanisms Reveals Layers of Complexity Complexity ---- a Case Studya Case Study

Added Citruline Should Increase Added Citruline Should Increase Arginine and Promote GrowthArginine and Promote Growth

Citruline Argininosuccinate Arginine Arg-t-RNA••• •••

Citruline0

FeedFeed--Forward Inhibition Causes Forward Inhibition Causes Self Starvation !Self Starvation !

Citruline Argininosuccinate Arginine Arg-t-RNA••• •••

Citruline0

But analysis shows that it cannot

Alternative Fates for Arginine Alternative Fates for Arginine Might Causes Self StarvationMight Causes Self Starvation

Citruline Argininosuccinate Arginine Arg-t-RNA••• •••

Citruline0

Design principle for control of branch points shows that it can, under certain circumstances

What Might be the Normal Function What Might be the Normal Function of Feedof Feed--Forward Inhibition?Forward Inhibition?

Xn-1 Xn Xn+1X3 •••X2X1X0

A well-controlled comparison shows that there is no significant difference, with or without feed-forward inhibition by the penultimate end product

What About Other FeedWhat About Other Feed--Forward Forward Inhibitors?Inhibitors?

Xn-1 Xn Xn+1X3 •••X2X1X0

Analysis shows that stability andtemporal responsiveness are optimal when X1 is the feed-forward inhibitor

( design)

Enzyme Complexes to Eliminate Enzyme Complexes to Eliminate Diffusion DelaysDiffusion Delays

Experimental evidence for the isoleucine biosyntheticpathway confirms the predicted design with the firstintermediate as feed-forward inhibitor and a complex involving the first and last enzyme of the pathway

Xn-1 Xn Xn+1

X2

•••

X3X1X0

X4Xn-2

( structure)

MotivationImportance of comparisonWell-controlled comparisons

MethodologyModeling strategiesCanonical nonlinear formalismMathematically controlled comparisons

Biological design principlesAnticipatory control in biosynthetic pathways

Coupling of elementary gene circuitsDemand for alternative modes of gene control

OutlineOutline

Two Extreme Forms of Coupling Two Extreme Forms of Coupling Gene ExpressionGene Expression

1mRNA, XNA, X 4

AA, X5

3Inducer, X6Substrate, X

2Enzyme/Regulator, X X0/

Perfect Coupling

1mRNA, X

2Enzyme, X

3Inducer, X

0Regulator, X

6Substrate, X

NA, X 4

AA, X5

Complete Uncoupling

( information)

EquationsEquationsPerfect Coupling Complete Uncoupling

dX1

dt= α1B − β1X1 X3 < X3L

dX1

dt= α1

pX2g12

pX3

g13p

− β1X1 X3L < X3 < X3H

dX1

dt= α1M − β1X1 X3H < X3

dX2

dt= α 2X1 − β2X2

dX3

dt= α3X2

g32 X4g34 − β3X2

h32 X3h33

dX1

dt= α1B − β1X1 X3 < X3L

dX1

dt= α1

uX3g13

u− β1X1 X3L < X3 < X3H

dX1

dt= α1M − β1X1 X3H < X3

dX2

dt= α 2X1 − β2X2

dX3

dt= α3X2

g32 X4g34 − β3X2

h32 X3h33

Constraints for External EquivalenceConstraints for External EquivalenceL

og

(en

zym

e co

nce

ntr

atio

n)

GainCapacity

Threshold

Log(inducer concentration)

Induction characteristic

α1u = β1

α1p

β1

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

α2

β2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

g12p /h22⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

h11h22

h11h22 −g12pg21

g13u = g13

p h11h22

h11h22 − g12pg21

Unique parameters

Design SpaceDesign Space

g13 =h11h22h33L24

g21g34

−h33L24

g34

g12

Line of equivalence

Regulatorg0

UnstableStable

•

••

• •

InducergHigh

Intermediate

Low

•

Repressor control

Regulatorg0

UnstableStable

• •

•

•

•

Inducerg

•

Low

High

Activator control

Example of Analytical ComparisonExample of Analytical Comparison

Robustness measured by parameter sensitivities

Parameter sensitivities defined as S(Vi , pj) =∂Vi∂pj

pjVi

S V3,β2( )p

S V3,β2( )u=

h11h22

h11h22 − g12pg21

< 1 for g12p < 0Ratio for comparison

External equivalence implies g13u = g13

p h11h22

h11h22 − g12pg21

Conclusion: Perfectly coupled circuit with repressor control is more robust than the equivalentcompletely uncoupled circuit

Savageau, Nature 229: 542 (1971)Becskei & Serrano, Nature 405: 590 (2000)

Response TimeResponse Time

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400 500 600

Time (min)

Flux

to p

rodu

ct V

-3- 2

0

0.5g12

Savageau, Nature 252: 546 (1974)Rosenfeld, et al., J. Mol. Biol. 323: 785 (2002)

Coupling of Gene Expression in Coupling of Gene Expression in Elementary CircuitsElementary Circuits

A

B

NA mRNA mRNA NA

Enzyme AA

Substrate Inducer

Regulator AA

C

Induction

Log

[Enz

yme]

Log [Substrate]

Log

[Reg

ulat

or] Directly Coupled

Inversely Coupled

Uncoupled

Log [Substrate]

Predicted Coupling of Gene Expression Predicted Coupling of Gene Expression in Elementary Circuitsin Elementary Circuits

Mode Capacity Predicted couplingPositive Small Inversely coupledPositive Large Directly coupledNegative Small Directly coupledNegative Large Inversely coupled

Experimental Evidence for Coupling of Experimental Evidence for Coupling of Gene Expression in Elementary CircuitsGene Expression in Elementary Circuits

Hlavacek & Savageau, J. Mol. Biol. 255: 121 (1996)

DUI

0 1 2 3 4 5-1

0

1

2

3

Log (Expression capacity of effector gene)

Log

(Exp

ress

ion

capa

city

of

regu

lato

r gen

e)

dsdC-dsdA

araC-araBADmetR-metE

lacI-lacZYA

hutIGC-hutUH

Wall, et al., Nature Rev. Genetics 5: 34 (2004)

MotivationImportance of comparisonWell-controlled comparisons

MethodologyModeling strategiesCanonical nonlinear formalismMathematically controlled comparisons

Biological design principlesAnticipatory control in biosynthetic pathwaysCoupling of elementary gene circuitsDemand for alternative modes of gene control

OutlineOutline

Dual Modes of Gene ControlDual Modes of Gene Control

( structure , design , information, physics, foundations)

Demand Theory of Gene ControlDemand Theory of Gene Control

A positive mode of control is predicted when there is a high demand for expression of a geneA negative mode of control is predicted when there is a low demand for expression of a gene

Savageau, PNAS 71:2453 (1974)

Life Cycle of Life Cycle of Escherichia coliEscherichia coli

L

H

Lactase LH

H HL L

DC (1- )D C

C

... ...

Time (hrs)

Region of Region of RealizabilityRealizability

Log

[C]

Log [D]

ModulatorPromoter

( design )

Rate and Extent of SelectionRate and Extent of Selection

H HL L

DC (1- )D C

C

... ...

Time (hrs)

C

B

LH

L

H

Lactase

A

D

E

F

D

Res

pons

e tim

e (h

rs)

ent o

f sel

ectio

n

mut

ant f

ract

ion)

Modulator

Promoter

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+2

1E+3

1E+4

1E+5

1E-7

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

1E+0

1E+2

1E+4

1E+6

1E+8

1E+10

C (h

rs)

Savageau (1998)

PredictionsPredictions

Cycling without colonization ≈ 26 hoursColonization without cycling ≈ 66 yearsRate of re-colonization ≈ 4 monthsEvolutionary response time ≈ 3 years

M. Savageau, Genetics 149:1677 (1998)

LYSOGENIC PHAGE λ

+Lytic LyticCRON CI

LYTIC PHAGE λ

Lytic LyticCRON CI ++

( design )

Sample of Design Principles Sample of Design Principles for Gene Circuitsfor Gene Circuits

Molecular mode control

Switching characteristics

Signaling Cross-talk

Coupling of expression

Alves & Savageau, Mol. Microbiol. 48:25 (2003)

Wall, et al., Nature Rev. Genetics 5: 34 (2004)

Savageau, Math. Biosci. 180:237 (2002)

Savageau, Genetics 149: 1677 (1998)

Comparisons are criticalNeed for well-controlled comparisonsExample illustrating the difficulties

MethodologyDistinguishing between two modeling strategiesCanonical nonlinear formalism has fundamental implicationsMathematically controlled comparisons are nearly ideal

Biological design principlesFeed-forward inhibition stabilizes long biosynthetic pathwaysRules for the coupling of expression in elementary gene circuitsDemand for gene expression provides the natural selection for alternative molecular modes of gene control

SummarySummary

AcknowledgementsAcknowledgements

Eberhard VoitDouglas IrvineRui AlvesGerald JacknowWilliam HlavacekMichael Wall

NSF, NIH, ONR, DOESloan FoundationPfizer