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 conditions heat 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
Mass/cell (µ doubling/hour) 200 min 20 min
2 µ Mass/cell (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)
2 1.5µ RNA/cell (µ doubling/hour) 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
Growth rate dependence of macromolecular composition for E. coli B/r [ Bremer & Dennis, 1996] doubling time minutes 100 60 40 30 24 20 Protein/mass 10 aa/od 460 5.8 5.5 5.1 4.8 4.5 4.0 RNA/mass 10 nuc/od 460 3.3 3.8 4.4 5.3 6.3 6.7 DNA/mass 10 8 gen/od 460 12.0 9.1 7.8 6.8 6.7 6.8 Cells/mass 10 8 cells/od 460 7.7 4.6 3.1 2.2 1.9 1.7 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 time minutes 100 60 40 30 24 20 RNA/cell 10 7 nuc 4.3 8.1 14.0 23.8 33.3 39.6 Ribosome/cell 10 3 7.9 15.3 26.1 44.3 61.8 72.5 r-protein/protein % 9 11.4 13.5 18.0 21.6 25 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: p m p m m mean-field description (rate eqn) m = # functional mrna/cell p = # of protein product/cell [p] = protein concentration d dt m = g m m m steady-state d p * dt p = p m p p = g m p m p V 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
Growth rate dependent expression of unregulated genes DNA replication C 40 min required to replicate chromosome fixed time of D 20 min between completion of one round of replication and cell division multiple replication forks for doubling time < C+D x oric terc gene copy # at position x on chromosome (different for plasmids) g x = 2 µ C (1 x)+ D [Cooper & Helmstetter, JMB 1968]
Transcription: determined by the abundance of free RNAP in exponential growth, RNAPs fall into one of 5 classes N tot = N ns + N r + N m + N free + N immat non-specifically bound to DNA N ns transcribing mrna. N m free N free immature N immat transcribing rrna, N r N tot directly measured N m,n r estimated from measured transcription rates partition inactive RNAPs into N ns,n free, N immat using microscopic model all needed parameters known except two (non-specific binding constant and RNAP maturation time) determine by fit of (N free +N immat )/N (fraction of cytoplasmic RNAP) to data from DNA-free minicells at different growth rates
Resulting partitioning of RNAP free RNAP rrn P2 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) P spc,p bla [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 ) [Bremer & Dennis, 1996] [Liang et al, J. Bact 2000] 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) independence of burstiness on growth rate despite strong growth-rate dependence of ribosome/mass global feedback mechanism (via mrna cleavage?)
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 expect stronger growth-rate dependence on plasmids [Lin-Chao & Bremer, MGG 1986]
Growth-rate dependence of regulated genes consider tsx init control only (for simplicity), with = c G 1 ln G A 1 ln G R slope n slope n f -1 f -1 ( ) n G R = 1 + f ( ) n G A = f 1 + [A]/K A 1 + [A]/K A K A ln([a]) K R 1 ([R]/K R ) n ( ) n 1 + [R]/K R ln([r]) -- 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 R E constitutive cooperative repression non-cooperative repression Negative control by an autorepressor R E wide occurrence in bacteria cooperative negative autoregulation can provide stable protein pools
Autoactivator A toggle switch A B region of bistability growth-rate dependent overlap of bistable region at slow and fast growth requires large fold-change and/or large cooperativity more complex behaviors for -- circuits involving srna -- circuits affecting growth
Back to the 2nd growth law 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 time minutes 100 60 40 30 24 20 RNA/cell 10 7 nuc 4.3 8.1 14.0 23.8 33.3 39.6 Ribosome/cell 10 3 7.9 15.3 26.1 44.3 61.8 72.5 r-protein/protein % 9 11.4 13.5 18.0 21.6 25 Rb synthesis/cell per minute 79 255 652 1476 2575 3625 # rrn gene/cell 12.4 16.5 22.0 27.6 32.9 37.5 Rb synthesis limited by rrna initiation max rrna transcription rate (100/min) x no. rrn genes/cell
Quantitative understanding of the 2nd growth law Ribosome availability determines the growth rate let = fraction of Rb synthesizing Rb and suppose ribosomes efficiently used in protein synthesis 1 d ln 2 dt M Rb = M Rb µm Rb 1 d Rb P 1- ln 2 dt M = (1 ) M µm P Rb P 1 : time for one Rb to synthesize a Rb µ = ln 2 7336 a.a./rb 5 min : master growth control 20 a.a./sec i is not observable, but can be deduced from Rb content: = M Rb M Rb + M P predicted growth rate dependence of Rb content: r M Rb M Rb + M P = µ /
Compare to data: r = µ / r = + µ / M RNA M Rb = r M tot (µ + ) 2 µ 5% 16 dbl/hr 22 a.a./sec no adjustable parameter! discrepancy suggests a fraction of inactive ribosomes model: a fraction of Rb inactivated then, µ = ( - ) Rb Rb 1- P or, r M Rb + M Rb M Rb + M Rb + M P = = µ +
Details more complicated:
Growth rate dependence may have several components: Divide proteome into 3 components: slow growth total protein ribosome related Q R P fast growth catabolic? Q R core (Q) P
Three component model of growth R: ribosomal proteins + affiliates; regulated by R (via ppgpp) Q: core proteins; regulated by Q (fixed, via autorepression?) Q R P: others; P = 1 - R - Q (not showing inactive ribosomes for clarity) 1 d ln 2 dt M = R R R M R µm R R 1 d P R P ln 2 dt M = P P R M R µm P Q Q 1 d ln 2 dt M Q = Q R M R µm Q P let M R = M Rb, then R = / µ = R / and R = M R M R + M P + M Q M Rb i relation between observables unchanged: r = µ / M R + M P + M Q i new constraint: r r max (1 Q )/ or µ µ max = r max i growth-rate dependent expression: P = 1 Q R µ max µ
include nutrient input (e.g., aa-limited growth) a.a. n flux balance at steady-state: a.a. consumption = a.a. supply R R P P E N Rb k E N E n n + K n = k E E N P n n + K n Q Q assume E P E: rate-limiting catabolic enzyme M R = (n, E,...) M P or R = P R + P + Q = 1 ( ) ( ) r = R / = r max + µ = r = r max + 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 : ( ) µ = r max r
Expt #1: Test the predicted form µ(, ) = r max Alter Rb elongation rate using antibiotic (Cm) (1 ) + K D [ Cm] KD [Harvey & Koch, 1980] test µ(, ) by measuring µ for different growth media ( ) at various Cm concentrations ( ) µ µ 0 1 0.8 0.6 0.4 µ 0 = 0.3 dbl/h µ 0 = 0.5 dbl/h µ 0 = 1dbl/h µ 0 = 2 dbl/h predicts stronger growth inhibition on faster growing cells 0.2 0 5 10 15 20 25 30 [ Cm]( µ M )
vary growth rate via N-source in M9 glycerol medium µ / µ 0 1 0.8 0.6 0.4 0.2 Alanine (µ 0 =0.35 dbl/hr) NH 4 + (µ 0 =0.51 dbl/hr) Cytidine (µ 0 =0.59 dbl/hr) NH 4 + (µ 0 =1.03 dbl/hr) + CAA 20 15 10 5 ( ) IC 50 µ M 0 5 10 15 20 25 30 [Cm] (in µm) 50 75 100 125 150 175 200 Doubling time (min) single parameter fit (K D ) to all data sets semi-quantitative agreement with prediction IC 50 strongly growth rate dependent K D
Expt #2: Test the predicted relation between µ and r i in the absence of drug, expect r = + µ / R % P P 30 25 20 15 10 5 14.6 dbl/h = 20 a.a./s NH 4 + +caa NH 4 + Cytidine Alanine Glucose 40 min (1.53 dbl/h) 70 min (.90 dbl/h) 81 min (0.74 dbl/h) 125 min (0.48 dbl/h) 58 min (1.04 dbl/h) 104 min (0.67 dbl/h) 130 min (0.46 dbl/h) 0.2 0.4 0.6 0.8 1 1.2 1.4 Growth Rate µ (dbl/h)
Expt #2: Test the predicted relation between µ and r i in the absence of drug, expect r = + µ / i with drug altering, expect µ = ( r max r) R % P P 30 25 20 15 10 5 14.6 dbl/h = 20 a.a./s NH 4 + +caa NH 4 + Cytidine Alanine Glucose 40 min (1.53 dbl/h) 70 min (.90 dbl/h) 81 min (0.74 dbl/h) 125 min (0.48 dbl/h) 58 min (1.04 dbl/h) 104 min (0.67 dbl/h) 130 min (0.46 dbl/h) 0.2 0.4 0.6 0.8 1 1.2 1.4 Growth Rate µ (dbl/h)
Expt #2: Test the predicted relation between µ and r i in the absence of drug, expect r = + µ / i with drug altering, expect µ = ( r max r) R % P P linearity of µ and r supports µ-independent suggests bottleneck of nutrient uptake reside in catabolic pathways (positive autoregulation) 30 25 20 15 10 5 14.6 dbl/h = 20 a.a./s R R Q NH 4 + +caa NH 4 + Cytidine Alanine a.a. P simple model of nutrient influx: n M P = k E E N P n + K n P Q Glucose 40 min (1.53 dbl/h) 70 min (.90 dbl/h) 81 min (0.74 dbl/h) 125 min (0.48 dbl/h) E n 58 min (1.04 dbl/h) 104 min (0.67 dbl/h) 130 min (0.46 dbl/h) 0.2 0.4 0.6 0.8 1 1.2 1.4 Growth Rate µ (dbl/h)
Expt #2: Test the predicted relation between µ and r i in the absence of drug, expect r = + µ / i with drug altering, expect µ = ( r max r) R % P P linearity of µ and r supports µ-independent suggests bottleneck of nutrient uptake reside in catabolic pathways (positive autoregulation) all minimal media exhibited similar r max ~ 25% composition of the metabolic core (Q)? 30 25 20 15 10 5 14.6 dbl/h = 20 a.a./s NH 4 + +caa NH 4 + Cytidine Alanine Glucose 40 min (1.53 dbl/h) 70 min (.90 dbl/h) 81 min (0.74 dbl/h) 125 min (0.48 dbl/h) 58 min (1.04 dbl/h) 104 min (0.67 dbl/h) 130 min (0.46 dbl/h) 0.2 0.4 0.6 0.8 1 1.2 1.4 Growth Rate µ (dbl/h)
Expt #2: Test the predicted relation between µ and r i in the absence of drug, expect r = + µ / i with drug altering, expect µ = ( r max r) linearity of µ and r supports µ-independent suggests bottleneck of nutrient uptake reside in catabolic pathways (positive autoregulation) all minimal media exhibited similar r max ~ 25% composition of the metabolic core (Q)? 40 Note: r max 40% for rich medium % r- Protein/Total Protein 35 30 25 20 15 10 5 2xYT 14.6 dbl/h = 20 a.a./s NH 4 + +caa NH 4 + Cytidine Alanine Glucose 40 min (1.53 dbl/h) 70 min (.90 dbl/h) 81 min (0.74 dbl/h) 125 min (0.48 dbl/h) 58 min (1.04 dbl/h) 104 min (0.67 dbl/h) 130 min (0.46 dbl/h) 0.5 1 1.5 2 2.5 3 Growth Rate µ (dbl/h)
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 Klumpp Matt Scott Funding: NIH, NSF, HFSP