Unraveling Bacterial Growth Laws · metabolism at high growth rates emerges as proteome-saving...

Unraveling Bacterial Growth LawsCoupling Energetics and Regulation in Cell Metabolism

Matteo Mori

Dottorato in Fisica XVII Ciclo

Supervisors: Prof. Enzo Marinari and Andrea De Martino

Roma – 29/10/2014

Bacteria: Life at its simplest

Rome – 29/10/2014 Matteo Mori – Unraveling Bacterial Growth Laws 2

Escherichia coli is the best known living organism, being extensively studies for 60+ years.

Engineered E. coli are used to produce many compounds, ranging from recombinant proteins (such as human insulin) to biofuels, on industrial scale.

The closest thing to a steady state for growing cells is exponential growth:

𝑁 𝑡 = 𝑁0𝑒𝜆𝑡 = 𝑁02

𝑡/𝜏𝐷

In this case, the averageproperties of the cellsare constant.

Metabolism

3

Survival of any cell relies on its ability to gather resources from the environments to grow(ultimately, to build a copy of itself).

Rome – 29/10/2014 Matteo Mori – Unraveling Bacterial Growth Laws

Metabolism

4

Survival of any cell relies on its ability to gather resources from the environments to grow(ultimately, to build a copy of itself).

This is a bit closer to the real thing.

E. coli core network (72 metabolites, 95 reactions) of central carbon metabolism

(Glycolysis, TCA cycle, penthose phosphate pathway, oxidativephosphorylation, plus a bunch of exchange reactions)

Full networks involve hundreds to thousands of reactions/metabolites, including compartimentalization(fundamental for eukaryotic cells,e.g. human)

Virtually any biological reaction iscatalyzed by enzymes.


Ribosomes

5

Translating bacterial ribosome.Artwork by Dale Muzzey

Roughly 60% of bacterial dry weight is made of proteins.

Proteins are produced by ribosomes, complexmacromolecules of protein and DNA.

Bacteria adapt their internalconstitution to the environment. About half of proteins in fast growing bacteria are ribosomalproteins.

PROTEINS

RNA

LIPIDSDNA


Growth laws and proteome partition model

6

A new framework offers a bird’s eye view on proteomepartitioning. For instance, let us focus on ribosomesand ribosome-affiliated proteins.

𝜙𝑅(𝜆) = 𝜙𝑅0 +𝜆

𝑘𝑅(general limitation)

𝜙𝑅 𝜆 = 𝜙𝑅𝑚𝑎𝑥 −

𝜆

𝑘𝐶(translational limitation)

Other proteome sectors can be defined based on generalized responses upon specific kind of growthlimitation (e.g. carbon, nitrogen).

A coarse grained model of proteome expression isjustified by the presence of a few signalling pathwaysregulating a large number of genes at once.

R

CQ

R

CQ

Nutrientlimitation

Inhibition of translation

Scott, Matthew, et al. "Interdependence of cell growth and gene expression: origins and consequences." Science 330.6007 (2010): 1099-1102.


How to model metabolism and gene regulations

7

Steady state Kinetic

Coarse grained

(a few proteome sectors with specific functions)

𝜙𝑅 = 𝜙𝑅0 +𝜆

𝑘𝑅

𝜙𝐶 = 𝜙𝐶0 +𝜆

𝑘𝐶𝜙𝐶 + 𝜙𝑅 + 𝜙𝑄 = 1

Genome scale

(about 1000 chemical reactionsinvolving 700 chemical species)



8


Coarse grained



𝑘𝑅



Genome scale


Still too hard, even for fixed proteinconcentrations!



9


Coarse grained



𝑘𝑅



Genome scale


Integration withFlux Balance Analysis (FBA)



Constraint-based modelling and Flux Balance Analysis

10

A kinetic model of metabolism is defined by a system of differential equations for the concentrations [𝑐𝜇]:

𝑑

𝑑𝑡𝑐𝜇 =

𝑖

𝑆𝜇𝑖𝑥𝑖

where the fluxes 𝑥𝑖 are complex (often unknown) functions of substrates and enzymeconcentrations. At steady state one can ignore concentrations and kinetics.

subject to

Typical objective function:biomass production (growth rate 𝜆)

Predictions on metabolite productions,maximum growth rate for a givenEnvironment, effects of gene deletions…

iii

i

ii

uxl

xS

0

)(max xcx

FBA

Constraints Optimization

Ort

h, J

effr

ey D

., In

es T

hie

le, a

nd

Ber

nh

ard

Ø. P

alss

on

. "W

hat

is f

lux

bal

ance

an

alys

is?.

" N

atu

re b

iote

chn

olo

gy

28

.3 (

20

10

): 2

45

-24

8.


Constrained Allocation Flux Balance Analysis (I)

11

1)()()(

0

QREC

iii

i

ii

xx

uxl

xS

)(max xcx

RRR

Ei

iiEE

CCCC

w

xw

xw

0

0

0

||

subject to

CA

FBA

C-sector: catabolic proteins. Proportional to carbon nutrientuptake.E-sector: enzymes (all internal reactions)R-sector: ribosome-affiliated proteins.Q-sector: fixed "housekeeping" sector.

Matteo Mori, Andrea De Martino, Enzo Marinari, and Terence Hwa"Constrained Allocation Flux Balance Analysis". Submitted to Molecular Systems Biology


Constrained Allocation Flux Balance Analysis (II)

12

Predictions from the coarse grained model are reproduced. Furthermore, CAFBA provides detailedpredictions aboutindividual fluxes.

In particular, anaerobicmetabolism at high growthrates emerges asproteome-saving strategyat high growth rates, coherently with evidencesof active regulation.


13

Constrained Allocation Flux Balance Analysis (III)

Randomization procedure. For instance:

𝑝 log10𝑤𝑖 =1

𝛿for 𝑤 −

𝛿

2≤ 𝑤𝑖 ≤ 𝑤 +

𝛿

2

Best choice: 𝛿 = 1. Many interesting results

• TCA/glycolysis switch predicted at high growth rates

• EMP/ED pathway switch

• NAD(P) transhydrogenase activity

• Quantitative agreement with exp data: acetate excretion, growth yield (Vemuri 2006, Basan 2014)



14


Coarse grained



𝑘𝑅



Genome scale






15


Coarse grained



𝑘𝑅


𝑘𝐶𝜙𝐶 + 𝜙𝑅 + 𝜙𝑄 = 1 Role of the offsets (𝜙𝑅

0 and 𝜙𝐶0)

in dynamical environments

Genome scale





Optimality of the proteome growth laws

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Offsets in the proteome growth laws are clearly suboptimal. In fact:

𝜙𝑅 = 𝝓𝑹𝟎 +

𝜆

𝑘𝑅, 𝜙𝐶 = 𝝓𝑪

𝟎 +𝜆

𝑘𝐶, 𝜙𝐶 + 𝜙𝑅 + 𝜙𝑄 = 1

Solving for growth rate:

𝜆 = 𝟏 − 𝝓𝑸 −𝝓𝑹𝟎 −𝝓𝑪

𝟎 𝑘𝐶𝑘𝑅

𝑘𝐶+𝑘𝑅.

Why are the offsets larger than zero? Physical constraints may be at work.For instance, ternary complexes (tRNA) – ribosome binding affinity is limited by diffusion of TC ("molecular crowding"), implying 𝜙𝑅

0 ≥ 0.Or… can it be done on purpose?

Klumpp, S. et al., "Molecular crowdinglimits translation and cell growth."PNAS 110: 16754-16759 (2013).


Upshifts

17

Interesting experiment: prepare a colonyin exponential phase and suddenly changethe medium they grow in.Production rates vary in a few minutes, butnot the instantaneous growth rate:

)0()0(

0

11

0

)(

ff

R

i

R

t

i

R

f

R

f

i

t

e

t

tf

Brunschede, H., T. L. Dove, and H. Bremer. "Establishment of exponential growth after a nutritional shift-up in Escherichia coli B/r: accumulation of deoxyribonucleic acid, ribonucleicacid, and protein." Journal of bacteriology 129.2 (1977): 1020-1033.

We only need the steady state ribosomal proteomefractions and growth rates to fully describe the upshift.

The offset 𝜙𝑅0 allows to bypass slow growth phases!


The offset is related to the steady-state/upshift tradeoff

18

Fitting upshift data allows to extract offsetsestimates for all proteome sectors activelyused for growth (not the "housekeeping" fraction).

About a 30% of total proteome is activelyused during upshifts!

Optimality can be studied introducing a fitness (total mass increase during upshifttime). Results consistent with adaptation.

max0 / RR

David Erickson, Matteo Mori, Severin Schink,"Investing in a proteome stock as strategy to cope with environmental changes."

In preparation for Nature Communications

Time after upshift (h)

Gro

wth

Rat

e (1

/h)

-1 0 1 2

00

.30

.60

.9


19

Conclusions

Summary

o A genome-scale model of intermediate metabolism can be integratedwith simple phenomenological laws to obtain striking agreement with both flux and proteome data.

o Proteome growth laws can be studied in the light of the biologicallymeaningful scenario of nutritional shifts. Apparent inefficiencies in proteome allocation can be the natural result of adaptation to dynamicenvironments.

Take-home message

Mind the interplays between

o Statics / Dynamicso Coarse grained / Large scale modelingo Physical constraints / Optimization


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Other works

o Thermodynamics constrains the space of feasible fluxes in FBA. Fluxloops can be identified and (sometimes) removed from a given fluxconfiguration. In particular, identification of flux loops can be difficult in large networks. Monte Carlo algorithms can be used in such cases.

De Martino, Daniele, Fabrizio Capuani, Matteo Mori, Andrea De Martino, and Enzo Marinari."Counting and correcting thermodynamically infeasible flux cycles in genome-scale metabolic networks."

Metabolites 3, no. 4 (2013): 946-966.

o What if there is no obvious flux to be optimized? One possibility isuniform sampling of the flux space, using Hit-and-Run Markov Chain Monte Carlo. This is not a trivial task, due to high dimensionality and scale hetereogeneity.

De Martino, Daniele, Matteo Mori, and Valerio Parisi."Uniform sampling of steady states in metabolic networks: heterogeneous scales and rounding".

Submitted to PLoS ONE. arXiv:1312.5228 [cond-mat.stat-mech]


Unraveling Bacterial Growth Laws · metabolism at high growth rates emerges as proteome-saving...

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