Transcript E. coli

Lecture #7
Estimation and Orders of Magnitude
Estimation
Orders of Magnitude
• Powers of 10:
http://micro.magnet.fsu.edu/primer/java/s
cienceopticsu/powersof10/
• Cell size and scale:
http://learn.genetics.utah.edu/content/beg
in/cells/scale/
Content
1.
Some Overall Observations
2.
Metabolism
I.
What are Typical Concentrations?
II.
What are Typical Metabolic Fluxes?
III. What are Typical Turnover Times?
IV. What are Typical Power Densities?
3.
Macromolecules
I.
What are Typical Characteristics of a Genome?
II.
What are Typical Protein Concentrations?
III. What are Typical Fluxes?
IV. What are Typical Turnover Times?
4.
Cell Growth and Phenotypic Functions
5.
Summary
Key Concepts
• Characteristic orders of magnitude for key
quantities that characterize cellular functions
can be estimated
• Data on cell size, mass, composition, metabolic
complexity, and genetic makeup are available
• Numerous databases now available on the web
• Useful estimates of fluxes, concentrations,
kinetics, and power densities in the
intracellular environment can be made based
on this data
Enrico Fermi (1901 1954) was an Italian
physicist, particularly
remembered for his work
on the development of
the first nuclear reactor,
and for his contributions
to the development of
quantum theory, nuclear
and particle physics, and
statistical mechanics.
Famous for quick
answers through
back-of-theenvelope
calculations
Introduction to Fermi problems
• The classic Fermi problem is:
"How many piano tuners are
there in Chicago?"
One approximation…
•
•
•
•
•
•
•
•
•
•
Thzere are approximately 5,000,000 people living in Chicago.
On average, there are two persons in each household in Chicago.
Roughly one household in twenty has a piano that is tuned regularly.
Pianos that are tuned regularly are tuned on average about once per year.
It takes a piano tuner about two hours to tune a piano, including travel time.
Each piano tuner works eight hours in a day, five days in a week, and 50 weeks
in a year.
From these assumptions we can compute that the number of piano tunings in a
single year in Chicago is
(5,000,000 persons in Chicago) / (2 persons/household) × (1 piano/20
households) × (1 piano tuning per piano per year) = 125,000 piano tunings
per year in Chicago. We can similarly calculate that the average piano tuner
performs
(50 weeks/year)×(5 days/week)×(8 hours/day)/(1 piano tuning per 2 hours
per piano tuner) = 1000 piano tunings per year per piano tuner. Dividing gives
(125,000 piano tuning per year in Chicago) / (1000 piano tunings per year per
piano tuner) = 125 piano tuners in Chicago.
Real significance …
• Possible to estimate key biological
quantities on the basis of a few foundational
facts and simple ideas from physics and
chemistry.
• Numbers collected by the scientific
community that initially appear unrelated
are brought together as a tool of inference to
shed light on biological mechanisms.
Biological examples
• How many proteins can be produced from a
single mRNA in E. coli?
• How many ATP synthase complexes are
required for optimal growth on glucose in E.
coli?
proteins/mRNA: method 1
• RNA nucleotide residues / cell: 7.3*107
• Amino acid residues / cell: 8.7*108
– Source: Neidhardt (Vol. 2/Table 2/pg. 1556)
• Fraction of RNA that is mRNA: 0.03 – 0.05
– Source: PMID 11713332
• Total mRNA nucleotide residues: 2,190,000 –
3,650,000 nt
• Average length of mRNA: 1,100 nt
• Number of mRNA / cell: 2000-3300
• Average length of protein: 367 AA
• Number of proteins / cell: 2.4 million
• 725-1200 proteins / mRNA:
proteins/mRNA: method 2
• Average length of mRNA: 1,100 nt
• A ribosome can bind every: 50 nt (structural
consideration)
• Maximum ribosome loading: 22 ribosomes/transcript
• Rate of translation: 16 AA / sec
• All ribosomes working together: 352 AA / sec
• Average length of protein: 367 AA
• Effective translation speed: About 1 protein/sec
• Average half-life of mRNA: 6 minutes (360 seconds)
• Mean lifetime of mRNA = 519 seconds (half-life / ln2)
• 519 proteins/mRNA
Let’s see how we did…
Marcotte et al., NBT 2007
Biological significance:
• Many expressed
genes in bacteria are
transcribed only
once per cell cycle
• Some cells fail to
produce an essential
message during a
cycle, and so must
depend on existing
messages and/or
proteins for survival
Another example: ATP synthase
• Motivation: membrane proteins notoriously difficult to quantify
• Maximum velocity of ATP synthase: 230 revolutions / sec
(828,000 / hr) [PMID 15668386]
• 3 ATP produced / revolution
• 2.5 million ATP / hr synthase
• Modeled flux required through ATP synthase: 52.0479
mmol/gDwh
– Input: Aerobic + 10 mmol glucose / gDwh
• With 2.8*10-13 gDw/cell, and using Avogadro’s number  Need
8,773,194,024 ATP / hr to grow optimally [growth rate of 0.7367
doublings/hr or a doubling time of about 1 hr]
• Need 3509 ATP synthase complexes working at Vmax
• Number of inner membrane proteins is 200,000
• Each ATP synthase complex has 22 proteins
• ATP synthase takes accounts for 40% of inner membrane
proteins (constraint for a future genome-scale model?)
Resource: BioNumbers database
Species
# BioNumbers
E. coli
920
H. sapiens
667
S. cerevisiae
394
Source: http://bionumbers.hms.harvard.edu/
BioNumbers is coordinated and developed by Ron Milo at the Weizmann Institute in Israel.
Orders of Magnitude
SOME OVERALL OBSERVATIONS
The Interior of a Cell:
a crowded place
Courtesy of David Goodsell
http://mgl.scripps.edu/people/goodsell/
The Cellular Environment:
highly organized in space (and time)
Typical Cellular Composition
Cellular Composition:
historic E. coli data
Representative Time Scales
Multi-scale
relationships:
metabolism,
transcription,
translation,
phenotypes
Small molecule scale
METABOLISM
The compounds
WHAT ARE TYPICAL METABOLITE
CONCENTRATIONS?
Typical Metabolite Concentration
1. The number of different metabolites present in E. coli is on the order of 1000.
2. An average metabolite has a median molecular weight of about 312 gram/mol.
3. We estimate the typical metabolite concentration:
and:
A typical metabolite concentration translates into about:
19,000 molecules per cubic micron!
Intracellular metabolite concentrations in glucose-fed,
exponentially growing E. coli
Rabinowitz et al.
Nature Chemical Biology (2009)
Intracellular metabolite concentrations in
glucose-fed, exponentially growing E. coli
Rabinowitz et al.
Nature Chemical
Biology (2009)
Size Distribution of Metabolites
Publicly Available Metabolic
Resources
Reaction rates
WHAT ARE TYPICAL METABOLIC
FLUXES?
What are Typical Turnover
Times?
Reaction versus Diffusion
1. Rate of diffusion varies with many chemical parameters
2. Estimating maximal reaction rates:
One million molecules per cubic micron (cell) per second!
Turnover Times of Glucose in E. coli
Estimating a glycolytic flux
1. The total stoichiometric amount of
glucose that is needed to generate
one E. coli cell is about 3 billion
molecules per cell.
2. Doubling time for E. coli is 60 min.
3. Volume of the E. coli cell is 1-2µm3
Glucose turnover in rapidly growing E. coli:
•
Extracellular Glucose concentration: 1-5 mM (6-30 x 105 molecules/cell)
•
Turnover time is on the order of sec
Turnover times in RBC glycolysis
Fast
and slow:
Distributed
time
constants
The Measured Time Response of
the Energy Charge
(2ATP+ADP)
2(ATP+ADP+AMP)
A bi-phasic response:
rapid decay and slow recovery
TWO FUNDAMENTAL CONTROL/REGULATORY CHALLENGES:
1. “Disturbance rejection” – return to the original state
2. “Servo” – transition from one steady state to the other steady state
The rapid response of energy
transducing membranes
(Redox Metabolism)
Charge on Energy Transducing
Membranes
• Majority of biological
energy transducing
membranes have
potential between -180
and -230 mV
• Bi-lipid layers become
physically unstable at 280 mV
Magnitude of the potential gradient
1. As presented above the potential is on the order
of -220-240 mV across the energy transducing
membrane.
2. The thickness of the lipid bi-layer is on the order
of 7nm.
3. So the potential gradient across this membrane is:
• 230 mV/7 nm = 300,000 V/cm
4. A potential gradient of 1,000 V/cm produces a
spark in the air (car spark plug).
ESTIMATING THE NUMERICAL
VALUE OF KINETIC CONSTANTS
Kinetic Constants of E. coli Enzymes
32 mM
• Majority of kinetic
information is based on
the in vitro measurements
– might not be
physiologically relevant
•Average Enzyme
concentration s on the
order of an average kinetic
constant (S ~ Km)
http://www.brenda-enzymes.info/
Typical Enzyme Turnover Times
1 min
http://www.brenda-enzymes.info/
The Distributions of Gibbs Free
Energies in iAF1260
Exothermic
Endothermic
WHAT ARE TYPICAL POWER
DENSITIES?
1. Power output of rat mitochondria
• Typical ATP production in mitochondria is
6 x 10-19 mol ATP/mitochondria/sec.
• Volume of the inner matrix in mitochondria is 0.27 μm3
• The energy of the phosphate bond is about 52 kJ/mol ATP
6 1019 molATP mitochondria
52kJ
13
3
3



1
.
1

10
W
/
m
m

0
.
1
pW
/
m
m
mitochondria  sec
0.27mm3
molATP
2. Power output of chloroplast in C. reinhardtii (green algae)
• Typical ATP production in chloroplast:
9.0 x 10-17 to 1.4 x 10-16 mol ATP/chloroplast/sec.
• Volume of a chloroplast 17.4 μm3
9  10 17  1  10 16 molATP chloroplas t
52kJ
13
3
3



2

10
w
/
m
m

0
.
2
pW
/
m
m
chloroplas t  sec
molATP
17.4mm 3
3. Power production density in a rapidly growing E. coli
• ATP production: 0.3 - 2.0 x 10-17 mol ATP/cell/sec
• Volume of E. coli 1 μm3
0.3  2.0 10 17 molATP cell
52 kJ
13
3
3



2
.
0

10

10
W
/
m
m

0
.
6
pW
/
m
m
cell  sec
1mm3 molATP
4. Power production by the sun
• Radiant power of the sun 3.86 x 1026 W
• Volume of the sun is 1.4 x 1027 m3
3.86 10 26W
1.412 10 27 m 3
 0.27
W
m3
 2.7 10 19W / mm 3  0.3 10 6 pW / mm 3
The power density of the sun is six orders of magnitude lower
Summary: metabolism
• Diffusion times are 1-10 msec faster than reactions
• Average concentration is about 30 mM
• Maximal fluxes are about a million molecules per m3
per sec
• Redox pools respond on the order of sec or faster,
energy charge on the order of a min
• Average Km is 32 mm close to substrate
concentrations
• Enzyme turnover times are < min
• Power densities are on the order of 0.1-0.5 pW/m3
Macromolecular scale
SYNTHESIS OF MACROMOLECULES:
DNA, RNA AND PROTEIN
Characteristics
of Genomes
- First sequenced genome (1995)
- Smallest free living organism
Features of the E. coli Genome
rRNA & tRNA
Features of the Human Genome
Topic
Total size of the genome:
Percentage of adenine (A) in the genome:
Percentage of cytosine (C) in the genome:
Percentage of bases not yet determined:
Highest gene-dense chromosome:
Least gene-dense chromosomes:
Percentage of DNA spanned by genes:
Percentage of exons:
Percentage of introns:
Percentage of intergenic DNA:
The average size of a gene:
The longest gene:
Average length of an intron:
Most common length of an intron:
Occurrence rate of SNPs:
SNRs:
Occurrence rate of genes:
RNA genes:
Statistic
approximately 3,200,000,000 bp
54%
38%
9%
chromosome 19 with 23 genes per 1,000,000 bp
chromosome 13 and Y with 5 genes per 1,000,000 bp
between 25% and 38%
1.1 to 1.4%
24% to 37%
74% to 64%
27,000 bp
dystrophin (a muscle protein) with 2,400,000 bp
3,300 bp
87 bp
roughly 1 per 1,500 bp
12,228,116
about 12 per 1,000,000 bp
4,150
Based on NCBI assembly Build 36 (released 2005) (http://www.ensembl.org/Homo_sapiens/index.html)
WHAT ARE TYPICAL PROTEIN
CONCENTRATIONS?
Protein Concentration in E. coli
1. Cells represent a fairly dense solutions of proteins
2. Concentration of total protein in cells falls in the range: 200 – 400 mg/ml
3. For E. coli we can assume:
• A cell has 1000 or so different proteins expressed at significant levels
• Average molecular weight of a protein is: 35 kDa.
• Protein is about 15% of wet weight of the cell or about 55% of the dry
cell weight
About 2500 molecules of a particular protein molecule per cubic micron!
With 1000 proteins present in the cell the total amount of protein molecules
is: 2.5 x 106 proteins/cell
Size distribution of ORF or Protein sizes
in E. coli
Distribution of Protein Concentrations
in E. coli
Size distribution of protein concentrations in E. coli K12 MG1655. Panel A: Relative
log (base 2) values of protein abundances rank-ordered; Panel B: Relative protein
abundance distribution.
Publicly Available Proteomic Resources
WHAT ARE TYPICAL SYNTHETIC
FLUXES OF MACROMOLECULES?
Typical Fluxes: DNA synthesis
1. The E. coli genome can be replicated in 40 min with 2
replication forks – the rate of DNA polymerase is:
2. The rate of RNA polymerase is much slower:
Process
Rate
DNA Replication (DNA -> DNA) 900 bp/sec/fork
Transcription (DNA -> RNA) 40-50 bp/sec
Translation (RNA -> Protein 12-20 amino acids/sec
Protein Synthesis in E. coli
1.
The rate of the ribosome is on the order of 12-21 peptide
bonds/ribosome/sec in rapidly growing E. coli.
2.
The amount of ribosomes present in E. coli depends vastly on the growth rate:
on the order of: 7x103 – 7x104 ribosomes/cell
3.
The total amount of peptide bonds that are formed in E. coli as a function of
growth rate can be estimated:
12  21 pb
7 x10 3  7 x10 4 ribosomes 8 x10 4  1.5 x10 6 pb


ribosome  sec
cell
cell  sec
4.
This value is equivalent to:
300  900 proteins 1  3x10 6 proteins

cell  sec
cell  hour
Protein Synthesis in Mammalian cell
1. The total amount of mRNA from a single gene in the
cytoplasm of the murine cell is on the order of 40,000
mRNAs/cell
2. The rate of the ribosome is 20 peptide bonds/cell/sec
3. The ribosomal spacing is 90-100 nucleotides/mRNA
4. This leads to the protein production rate in murine cell:
3000  6000 proteins 1  2 x10 7 proteins

cell  sec
cell  hour
The whole-cell scale
CELL GROWTH AND PHENOTYPIC
FUNCTIONS
Phenotypic characteristics of E. coli:
Aerobic (60 min) and Anaerobic profile (90 min)
Flux Name
Glucose Uptake
Rate (mmol/gDW/h)
anoxic
oxic
9.02 +/- 0.23
17.9 +/- 1.2
molecules/µm3/sec
oxic
anoxic
9.0 x 105
0
4.50 x 105
7.90 x 10 5
1.76 x 105
1.70 x 105
0
Oxygen Uptake
Formate Secretion
15.8 +/- 1.8
14.92+/-0.21
3.51 +/- 0.47
Acetate Secretion
10.9 +/- 0.8
3.37 +/- 0.02
5.45 x 10 5
Ethanol Secretion
7.4 +/- 0.6
0
Succinate Secretion
1.1+/- 0.4
0
3.70 x 10 5
5.50 x 10 4
Lactate Secretion
0.2 +/- 0.1
0
1.0 x 104
0
7.50 x 105
0
0
Synthesis of an E. coli Cell:
order-of-magnitude estimation of fluxes
•
•
There are 3.0 x106 proteins per cell, each with an average length of 316 AA.
If the ribosome can make 20 peptide bonds/sec = 1200 pb/min = 72,000
pb/hr:
Nucleotide requirement per hour (or cell division):
RNA:
Stable RNA
mRNA
DNA:
chromosome
for a grand total of approximately 9.26 x 107
nucleotides/cell for synthesis of RNA and DNA
molecules for one cell.
Synthesis of an E. coli Cell:
order-of-magnitude estimation of fluxes (cont)
•
The glucose uptake has to be balanced for energy production rate (at
about 18 ATP/glucose-aerobically and 3 ATP/glucose-anaerobically) and
to meet the biosynthetic rates, that will also have to include cell wall and
lipid synthesis.
•
Thus the energy equivalent produced is:
-230 mV -> 105 V/cm
Energy production: 0.2 – 1.0
METABOLISM
ATP Production rate:
Glycolytic flux: 3 x 109 molecules/cell/h
0.8 - 4 x 1010 molecules/cell/h
pW/µm3
H+
ADP
ATP
H+
Protein
DNA replication rate:
Nucleotide Flux:
900 bp/sec/fork
5 x 108 nucleotides/cell/h
Amino Acid Flux:
9 x 108 amino acid/cell/h
DNA
Protein production rate:
tRNA
3 x 106 proteins/cell/h
Fraction of RNAP
synthesizing tRNA/rRNA:
0.28-0.77
RNA Polymerase rate:
5 x 108 nucleotides/cell/h
Cell doubling time:
mRNA
Ribosome rate: 3 x 109 peptide bonds/cell/h
60 min
Overall metabolic rates in E. coli:
Implications for bioprocessing
Flux Name
Glucose Uptake
Formate Secretion
Acetate Secretion
Ethanol Secretion
Succinate Secretion
Lactate Secretion
Rate (mmol/gDW/h) molecules/µm3/sec
17.9 +/- 1.2
2.98E+06
15.8 +/- 1.8
2.63E+06
10.9 +/- 0.8
1.83E+06
7.4 +/- 0.6
1.25E+06
1.1+/- 0.4
1.83E+05
0.2 +/- 0.1
3.33E+04
• Reduced by-products are produced anaerobically
• Glycolytic flux often is the entry point of the sugar to the
metabolism
• E. coli is a commonly used for metabolic engineering
applications
• Successful metabolic engineering design is usually
characterized by its volumetric productivity
Limits on Volumetric Productivity
• Anaerobically E. coli has substrate uptake rate (SUR) of:
15 – 20 mmol Glucose/ gDW/h
• Which translates to:
1.5 gram Glucose /L/h
• If all the glucose is converted to the desired product (i.e.
D-Lactate), the VOLUMETRIC PRODUCTIVITY of this strain
design is:
~ 3 gram Lactate/L/h
• Some metabolically engineered E. coli strains have SUR
higher then reported above, leading to higher volumetric
productivity.
FROM BACTERIA TO MAMMALS
Metabolic rate of major organs
Size range of living organisms
Figure taken: K. Schmidt-Nielsen, “Why is animal size so important”, 1984
Metabolic rate and body size
Figure taken: K. Schmidt-Nielsen, “Why is animal size so important”, 1984
Summary
• The size of a bacterial cell is around 1 µm with a weight of 1 pg.
• The interior of the cell is a viscous solution crowded with several
molecular species
• The cells are mostly composed of water and macromolecules with
simple metabolites forming only a small fraction.
• Typical concentrations of metabolites and enzymes within the cell fall
in the micromolar range with a wide distribution around the mean.
• Metabolites are present at an average concentration of 19,000
molecules/µm3, while enzymes have an average concentration of 2000
molecules/µm3.
• Diffusional response times for bacteria, on the order of milliseconds,
are much faster than the metabolic dynamics. Spatial distributions can
therefore be neglected.
• Metabolic fluxes occur at average rates of 104 to 105 molecules/µm3
/sec.
O-OF-MAGNITUDE: some examples
The magnitude of the bailout package
We'll start with a $100
dollar bill.
A packet of one hundred
$100 bills is less than 1/2"
thick and contains $10,000.
http://sketchup.google.com/
Believe it or not, this next
little pile is $1 million
dollars (100 packets of
$10,000).
$100 million is a little more
respectable. It fits neatly on
a standard pallet...
http://sketchup.google.com/
And $1 BILLION dollars... now we're really
getting somewhere...
http://sketchup.google.com/
Next we'll look at ONE TRILLION dollars. This is that
number we've been hearing about so much. What is a
trillion dollars? Well, it's a million million. It's a thousand
billion. It's a one followed by 12 zeros.
Ladies and gentlemen... I give you $1 trillion dollars...
So the next time you hear someone toss around the phrase "trillion dollars"... that's what
they're talking about.