Bacterial growth control & control by bacterial growth · • growth Carbon + Nitrogen biomass doubling time: 20 min to > 500 min depending on nutrient • survival – coping with

Bacterial growth control

& control by bacterial growth

PICB, Shanghai

July 18, 2008

Systems biology: from molecules to physiology

Physiology: the working of a living organism;

reproduction & adaptation to the environment

Bacterial physiology (E. coli):

bacteria can sense the environment

and rapidly adjust their “life style” • growth Carbon + Nitrogen biomass

doubling time: 20 min to > 500 min depending on nutrient

• survival

– coping with stressful conditionsheat shock, acid shock, osmotic response, oxidative stress, …

– non-growth conditions: stationary phase, dormancy, sporulation, …

This talk: Bacterial growth physiology• review of bacterial growth laws

• their impact on gene expression and genetic circuits

• phenomenological model of bacterial growth

Ma

ss/c

ell

(µ doubling/hour)

200 min 20 min

Ma

ss/c

ell

(µ doubling/hour)

200 min 20 min

• phenomenological law!

• dependence on the medium through growth rate µ only -- universal!

• cell mass (~size) increases exponentially with growth rate µ

• “natural” unit of growth rate ~ 1 doubling/hr (universal speed limit)

2µ

• RNA content increases more rapidly than cell mass (mostly protein)

macromolecular composition (e.g., RNA:protein) strongly µ dependent

similar growth laws seen in E. coli and other bacteria

RN

A/c

ell

(µ doubling/hour)

21.5µ

doubling timedoubling time minutesminutes 100100 6060 4040 3030 2424 2020

Protein/massProtein/mass 10101717 aa/OD aa/OD460460 5.8 5.8 5.55.5 5.15.1 4.84.8 4.54.5 4.04.0

RNA/mass RNA/mass 10101616 nuc/OD nuc/OD460 460 3.33.3 3.83.8 4.44.4 5.35.3 6.36.3 6.76.7

DNA/mass DNA/mass 101088 gen/OD gen/OD460460 12.012.0 9.19.1 7.87.8 6.86.8 6.76.7 6.86.8

Cells/massCells/mass 101088 cells/OD cells/OD460460 7.77.7 4.64.6 3.13.1 2.22.2 1.91.9 1.71.7

Growth rate dependence of macromolecular composition

for E. coli B/r [ Bremer & Dennis, 1996]

• protein cell mass

• strong increase in RNA/cell

• weak increase in DNA/cell

Growth-rate dependence

of the cellular RNA content

• cellular RNA ribosomal RNA

ribosome level from stoichiometry

• ribosome content growth rate dependent

• a significant fraction of cellular proteins

are ribosomal proteins at fast growth


RNA/cell RNA/cell 101077 nucnuc 4.34.3 8.18.1 14.014.0 23.823.8 33.333.3 39.639.6

Ribosome/cell Ribosome/cell 101033 7.97.9 15.315.3 26.126.1 44.344.3 61.861.8 72.572.5

r-protein/proteinr-protein/protein %% 99 11.411.4 13.513.5 18.018.0 21.621.6 2525

Alternative form of Schaecter et al’s 2nd growth law

What do the growth laws have to do with modern biology?

Consider constitutive gene expression:

mean-field description (rate eqn)

m = # functional mRNA/cell

p = # of protein product/cell

[p] = protein concentration

d

dtm = g m m m

d

dtp = p m p p

steady-statep* =

g m p

m pV

Q: growth rate dependence of [p*]?

-- many factors depend passively on growth rate: g, V, p, p (and others)

-- more complex for actively regulated genes

-- gene regulation mostly accompanied by changes in growth rate

growth rate dependence affects the robustness of genetic circuits

must consider growth dependence in physiological studies of gene reg

p

m

p

m

m

Growth rate dependent expression of unregulated genes

• DNA replication

– C 40 min required to replicate chromosome

– fixed time of D 20 min between completionof one round of replication and cell division

– multiple replication forks for doubling time < C+D

g

x= 2

µ C (1 x)+D

oriC

terC

x

[Cooper & Helmstetter, JMB 1968]

gene copy # at position x on chromosome

(different for plasmids)

• Transcription: determined by the abundance of free RNAP

transcribing rRNA, Nr

transcribing mRNA

. Nmnon-specifically

bound to DNA

Nns

immature

Nimmat

free

Nfree

in exponential growth, RNAPs fall into one of 5 classes

• Ntot directly measured

• Nm,Nr estimated from measured transcription rates

• partition inactive RNAPs into Nns,Nfree, Nimmat using microscopic model

• all needed parameters known except two

(non-specific binding constant and RNAP maturation time)

determine by fit of (Nfree+Nimmat)/N (fraction of cytoplasmic RNAP)

to data from DNA-free minicells at different growth rates

N

tot= N

ns+ N

r+ N

m+ N

free+ N

immat

Resulting partitioning of RNAP

• predicted growth-rate dependence of free

RNAP agree well with measured tsx rates for

several constitutive promoters

• rrn P2 not a constitutive promoter in

agreement with in vitro results (Gourse)

free RNAP

rrn P2

Pspc,Pbla

[Liang et al, JMB 1999]

• mRNA stability:

– lifetime lacZ mRNA (Liang et al. 1999)

1.9 min @ 0.6 dbl/hr, 2.4 min @ 3 dbl/hr

– genome-wide study (Bernstein et al. 2002):

small growth-rate dependence in mRNA lifetime

growth rate dependence of mRNA “levels”

• Growth-rate dependence of translation efficiency (“burstiness”)

independence of burstiness on growth rate

despite strong growth-rate dependence of ribosome/mass

global feedback mechanism (via mRNA cleavage?)

• avg burstiness = (total protein/cell • ln2/DT)/(measured total mRNA syn rate/cell)

• burstiness of r-protein mRNA (r-protein/cell • ln2/DT)/(tsx rate of Spc prom/cell)

• burstiness of lacZ mRNA ( -gal activity/total protein • ln2/DT)

• (total protein/total RNA) / (lacZ mRNA/total RNA)

[Bremer & Dennis, 1996]

[Liang et al, J. Bact 2000]

growth rate dependence of protein “levels”

Compare to experiments:-- generally good agreement for different promoters and genes

in different strains and grown in different media

• Expression from plasmids

-- gene dosage = plasmid copy number

-- pBR322: increases weakly with growth rate

[Lin-Chao & Bremer, MGG 1986]

expect stronger growth-rate dependence on plasmids

Growth-rate dependence of regulated genes

consider tsx init control only (for simplicity), with = c·G

GA =f 1

+ [A] / KA( )n

1+ [A] / KA( )n

GR =1+ f 1 [R] / KR( )

n

1+ [R] / KR( )n

ln GA

ln([A])

f -1

1

slope n

KA

ln GR

ln([R])

f -1

1

|slope| n

KR

-- growth-rate independent: fold-change (f) Hill coeff (n), dissoc const (K)

-- growth-rate dependent: transcription rate (c), cell volume, …

• Negative control by constitutively expressed repressor

ER

cooperative

repression

non-cooperative

repression

constitutive

• Negative control by an autorepressor

ER

wide occurrence in bacteria

cooperative negative autoregulation can provide stable protein pools

•Autoactivator

region of bistability growth-rate dependent

overlap of bistable region at slow and fast growth

requires large fold-change and/or large cooperativity

A

• toggle switch

BA

more complex behaviors for-- circuits involving sRNA

-- circuits affecting growth

• cellular RNA ribosomal RNA

ribosome level from stoichiometry

• ribosome content growth rate dependent

• a significant fraction of cellular proteins

are ribosomal proteins at fast growth


RNA/cell RNA/cell 101077 nucnuc 4.34.3 8.18.1 14.014.0 23.823.8 33.333.3 39.639.6

Ribosome/cell Ribosome/cell 101033 7.97.9 15.315.3 26.126.1 44.344.3 61.861.8 72.572.5

r-protein/proteinr-protein/protein %% 99 11.411.4 13.513.5 18.018.0 21.621.6 2525

Rb Rb synthesis/cellsynthesis/cell per minuteper minute 7979 255255 652652 14761476 25752575 36253625

# # rrn rrn gene/cellgene/cell 12.412.4 16.516.5 22.022.0 27.627.6 32.932.9 37.537.5

Rb synthesis limited by rRNA initiation

max rRNA transcription rate (100/min) x no. rrn genes/cell

Back to the 2nd growth law

• let = fraction of Rb synthesizing Rb

and suppose ribosomes efficiently used in protein synthesis

: master growth control

Quantitative understanding of the 2nd growth law

Ribosome availability determines the growth rate

i is not observable, but can be deduced from Rb content: =MRb

MRb + MP

PRb

1-1 : time for one Rb to synthesize a Rb

ln27336 a.a./Rb

20 a.a./sec5min

µ =

rMRb

MRb + MP

= µ /• predicted growth rate dependence of Rb content:

1

ln2

d

dtMRb = MRb µMRb

1

ln2

d

dtMP = (1 ) MRb µMP

MRNA MRb = r Mtot (µ + ) 2µ r = µ /

Compare to data:

• no adjustable parameter!

• discrepancy suggests a fraction of inactive ribosomes

model: a fraction of Rb inactivated

then, µ = ( - )

or, rMRb + MRb

MRb + MRb+ MP

= =µ+PRb

1- Rb

5%

16 dbl/hr

22 a.a./sec

r = + µ /

Details more complicated:

Growth rate dependence may have several components:

total protein

ribosome related

catabolic?

Divide proteome into 3 components:

slow growth

fast growth

R

P

Q

QR

Pcore (Q)

let MR = MRb, then R = /

Three component model of growth

µ = R / and R =MR

MR + MP + MQ

i relation between observables unchanged: rMRb

MR + MP + MQ

= µ /

1

ln2

d

dtMR = R RMR µMR

1

ln2

d

dtMP = P RMR µMP

1

ln2

d

dtMQ = Q RMR µMQ

QR

P

R: ribosomal proteins + affiliates; regulated by R (via ppGpp)

Q: core proteins; regulated by Q (fixed, via autorepression?)

P: others; P = 1 - R - Q

(not showing inactive ribosomes for clarity)

PR

R

P

QQ

i new constraint: r rmax (1 Q ) / or µ µmax = rmax

i growth-rate dependent expression: P = 1 Q R µmax µ

• flux balance at steady-state:

a.a. consumption = a.a. supply

kE NE

n

n + Kn

r = R / = rmax +( )

µ = r = rmax +( )

include nutrient input

(e.g., aa-limited growth)

E: rate-limiting catabolic enzyme

growth rate (µ) determined jointly by nutrient source/level ( )

and the speed of the ribosome ( )

simple Michaelis form if independent of µ (!)

can be tested by eliminating :

PR

na.a.

E

PR

QQ

NRb= kE E NP

n

n + Kn

assume E P

MR = (n, E ,...) MP

or R = P

R + P + Q = 1

µ = rmax r( )

Alter Rb elongation rate using antibiotic (Cm)

test µ( , ) by measuring µ

for different growth media ( )

at various Cm concentrations ( )

µ( , ) = rmax(1 )

+

predicts stronger growth inhibition

on faster growing cells[ ]( )Cm Mµ

00.3 dbl/hµ =

00.5 dbl/hµ =

02 dbl/hµ =

01dbl/hµ =

5 10 15 20 25 30

0.2

0.4

0.6

0.8

1

0

0

µ

µ

Expt #1: Test the predicted form

[ ]

D

Cm

KD

K

[Harvey & Koch, 1980]

vary growth rate via N-source in M9 glycerol medium

– single parameter fit (KD) to all data sets

– semi-quantitative agreement with prediction

– IC50 strongly growth rate dependent

µ / µ

0

[Cm] (in µM)

5 10 15 20 25 30

0.2

0.4

0.6

0.8

1

0

Alanine (µ0=0.35 dbl/hr)

NH4+ (µ0 =0.51 dbl/hr)

Cytidine (µ0 =0.59 dbl/hr)

NH4+ (µ0 =1.03 dbl/hr)

+ CAA

IC

50 µM( )

Doubling time (min)

50 75 100 125 150 175 200

5

10

15

20

DK

Expt #2: Test the predicted relation between µ and r

Growth Rate µ (dbl/h)

%P

R

P

0.2 0.4 0.6 0.8 1 1.2 1.4

5

10

15

20

25

30

i in the absence of drug, expect r = + µ /

14.6 dbl/h

= 20 a.a./s

130 min

(0.46 dbl/h)

125 min

(0.48 dbl/h)Alanine

81 min

(0.74 dbl/h)Cytidine

104 min

(0.67 dbl/h)

70 min

(.90 dbl/h)NH4

+

58 min

(1.04 dbl/h)

40 min

(1.53 dbl/h)NH4

++cAA

Glucose



%P

R

P

0.2 0.4 0.6 0.8 1 1.2 1.4

5

10

15

20

25

30

i with drug altering , expect µ = rmax r( ) i in the absence of drug, expect r = + µ /

14.6 dbl/h

= 20 a.a./s

130 min

(0.46 dbl/h)

125 min

(0.48 dbl/h)Alanine

81 min


104 min

(0.67 dbl/h)

70 min

(.90 dbl/h)NH4

+

58 min

(1.04 dbl/h)

40 min

(1.53 dbl/h)NH4

++cAA

Glucose



%P

R

P

0.2 0.4 0.6 0.8 1 1.2 1.4

5

10

15

20

25

30


linearity of µ and r supports µ-independent

suggests bottleneck of nutrient uptake reside

in catabolic pathways (positive autoregulation)

14.6 dbl/h

= 20 a.a./s

130 min

(0.46 dbl/h)

125 min

(0.48 dbl/h)Alanine

81 min


104 min

(0.67 dbl/h)

70 min

(.90 dbl/h)NH4

+

58 min

(1.04 dbl/h)

40 min

(1.53 dbl/h)NH4

++cAA

Glucose

simple model of nutrient influx:

MP = kE E NP

n

n + Kn

PR

na.a.

E

PR

QQ


14.6 dbl/h

= 20 a.a./s


%P

R

P

0.2 0.4 0.6 0.8 1 1.2 1.4

5

10

15

20

25

30

130 min

(0.46 dbl/h)

125 min

(0.48 dbl/h)Alanine

81 min


104 min

(0.67 dbl/h)

70 min

(.90 dbl/h)NH4

+

58 min

(1.04 dbl/h)

40 min

(1.53 dbl/h)NH4

++cAA

Glucose





all minimal media exhibited similar rmax~ 25%

composition of the metabolic core (Q)?




% r

- P

rote

in/T

ota

l P

rote

in

0.5 1 1.5 2 2.5 3

5

10

15

20

25

30

35

40 Note: rmax 40% for rich medium

2xYT

14.6 dbl/h

= 20 a.a./s130 min

(0.46 dbl/h)

125 min

(0.48 dbl/h)Alanine

81 min


104 min

(0.67 dbl/h)

70 min

(.90 dbl/h)NH4

+

58 min

(1.04 dbl/h)

40 min

(1.53 dbl/h)NH4

++cAA

Glucose




all minimal media exhibited similar rmax~ 25%

composition of the metabolic core (Q)?

Summary

• bacterial growth in different media

growth rate as a critical variable

• growth-rate dependent effects on gene expression

– complex dependence even for constitutive promoters

– growth-rate dependence can be minimized by negative autoregulation

– taming growth-rate dependent effects essential for

robust genetic circuits

• phenomenological model of growth rate control

– three components: R (ribosome+affiliates), Q (core), P (catabolic)

– experiment supports the simplest (coarse-grained) description of

metabolic control: trade-off between P and R

– core fraction similar for all minimal media tested

Stefan KlumppMatt Scott

Funding: NIH, NSF, HFSP

Documents

Bacterial growth control & control by bacterial growth · • growth Carbon + Nitrogen biomass doubling time: 20 min to > 500 min depending on nutrient • survival – coping with