bacterial growth control & control by bacterial growth · • growth carbon + nitrogen biomass...
TRANSCRIPT
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
doubling timedoubling time minutesminutes 100100 6060 4040 3030 2424 2020
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
doubling timedoubling time minutesminutes 100100 6060 4040 3030 2424 2020
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
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
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
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 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
(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
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 with drug altering , expect µ = rmax r( ) i in the absence of drug, expect r = + µ /
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
(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
simple model of nutrient influx:
MP = kE E NP
n
n + Kn
PR
na.a.
E
PR
Expt #2: Test the predicted relation between µ and r
14.6 dbl/h
= 20 a.a./s
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
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
i with drug altering , expect µ = rmax r( ) i in the absence of drug, expect r = + µ /
linearity of µ and r supports µ-independent
suggests bottleneck of nutrient uptake reside
in catabolic pathways (positive autoregulation)
all minimal media exhibited similar rmax~ 25%
composition of the metabolic core (Q)?
Expt #2: Test the predicted relation between µ and r
i with drug altering , expect µ = rmax r( ) i in the absence of drug, expect r = + µ /
Growth Rate µ (dbl/h)
% 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
(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
linearity of µ and r supports µ-independent
suggests bottleneck of nutrient uptake reside
in catabolic pathways (positive autoregulation)
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