1. Estimation and Orders of Magnitude
Lecture #7
Estimation and Orders of Magnitude

2. Estimation

3. Orders of Magnitude
Powers of 10:
Cell size and scale:

4. Content Some Overall Observations Metabolism
What are Typical Concentrations?
What are Typical Metabolic Fluxes?
What are Typical Turnover Times?
What are Typical Power Densities?
Macromolecules
What are Typical Characteristics of a Genome?
What are Typical Protein Concentrations?
What are Typical Fluxes?
Cell Growth and Phenotypic Functions
Summary

5. 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

6. Enrico Fermi ( ) 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-the-envelope calculations

7. Introduction to Fermi problems
The classic Fermi problem is: "How many piano tuners are there in Chicago?"

8. 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.

9. 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.

10. 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?

11. 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
Total mRNA nucleotide residues: 2,190,000 – 3,650,000 nt
Average length of mRNA: 1,100 nt
Number of mRNA / cell:
Average length of protein: 367 AA
Number of proteins / cell: 2.4 million
proteins / mRNA:

12. 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

13. Let’s see how we did… 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
Marcotte et al., NBT 2007

14. Another example: ATP synthase
Motivation: membrane proteins notoriously difficult to quantify
Maximum velocity of ATP synthase: 230 revolutions / sec (828,000 / hr) [PMID ]
3 ATP produced / revolution
2.5 million ATP / hr synthase
Modeled flux required through ATP synthase: 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 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?)

15. Resource: BioNumbers database
Species
# BioNumbers
E. coli
920
H. sapiens
667
S. cerevisiae
394
Source:
BioNumbers is coordinated and developed by Ron Milo at the Weizmann Institute in Israel.

16. SOME OVERALL OBSERVATIONS
Orders of Magnitude
SOME OVERALL OBSERVATIONS

17. The Interior of a Cell: a crowded place
Courtesy of David Goodsell

18. The Cellular Environment: highly organized in space (and time)

19. Typical Cellular Composition

20. Cellular Composition: historic E. coli data

21. Representative Time Scales

22. Multi-scale relationships: metabolism, transcription, translation, phenotypes

23. Multi-scale view of E. coli
colony
cell
nucleoid
macromolecule

24. Biological Scales and Systems Analysis
ecology
physiology
immunology
Molecular systems biology
Courtesy of Vito Quaranta, MD; Vanderbilt University, Nashville, TN

25. Small molecule scale
METABOLISM

26. WHAT ARE TYPICAL METABOLITE CONCENTRATIONS?
The compounds
WHAT ARE TYPICAL METABOLITE CONCENTRATIONS?

27. Typical Metabolite Concentration
The number of different metabolites present in E. coli is on the order of 1000.
An average metabolite has a median molecular weight of about 312 gram/mol.
We estimate the typical metabolite concentration:
and:
A typical metabolite concentration translates into about:
19,000 molecules per cubic micron!

28. Rabinowitz et al. Nature Chemical Biology (2009)
Intracellular metabolite concentrations in glucose-fed, exponentially growing E. coli
Rabinowitz et al. Nature Chemical Biology (2009)

29. Intracellular metabolite concentrations in glucose-fed, exponentially growing E. coli
Rabinowitz et al.
Nature Chemical Biology (2009)

30. Size Distribution of Metabolites

31. Publicly Available Metabolic Resources

32. WHAT ARE TYPICAL METABOLIC FLUXES?
Reaction rates
WHAT ARE TYPICAL METABOLIC FLUXES?

33. What are Typical Turnover Times?

34. Reaction versus Diffusion
Rate of diffusion varies with many chemical parameters
Estimating maximal reaction rates:
One million molecules per cubic micron (cell) per second!

35. Turnover Times of Glucose in E. coli
Estimating a glycolytic flux
The total stoichiometric amount of glucose that is needed to generate one E. coli cell is about 3 billion molecules per cell.
Doubling time for E. coli is 60 min.
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

36. Turnover times in RBC glycolysis
Fast
and slow:
Distributed time constants

37. 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:
“Disturbance rejection” – return to the original state
“Servo” – transition from one steady state to the other steady state

38. The rapid response of energy transducing membranes (Redox Metabolism)

39. 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

40. Magnitude of the potential gradient
As presented above the potential is on the order of mV across the energy transducing membrane.
The thickness of the lipid bi-layer is on the order of 7nm.
So the potential gradient across this membrane is:
230 mV/7 nm = 300,000 V/cm
A potential gradient of 1,000 V/cm produces a spark in the air (car spark plug).

41. ESTIMATING THE NUMERICAL VALUE OF KINETIC CONSTANTS

42. 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)

43. Typical Enzyme Turnover Times
1 min
‘fast’

44. The Distributions of Gibbs Free Energies in iAF1260
Exothermic
Endothermic

45. WHAT ARE TYPICAL POWER DENSITIES?

46. Power output of rat mitochondria
Typical ATP production in mitochondria is
6 x 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
Power output of chloroplast in C. reinhardtii (green algae)
Typical ATP production in chloroplast:
9.0 x to 1.4 x mol ATP/chloroplast/sec.
Volume of a chloroplast 17.4 μm3

47. Power production density in a rapidly growing E. coli
ATP production: x mol ATP/cell/sec
Volume of E. coli 1 μm3
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
The power density of the sun is six orders of magnitude lower

48. Power /Area – Heat transport
man-made power devices
Power /Area – Heat transport
nature-made power devices
Power /Volume – Power Density

49. Calculating power density of car engines
Common conversions: 1 hp = 746 W ; 1L = m3
The power density of E. coli is only two orders of magnitude lower then of MB McLaren

50. Power Densities
Specific power densities of biological and man-made energy generating machinery.

51. 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 pW/m3

52. SYNTHESIS OF MACROMOLECULES: DNA, RNA AND PROTEIN
Macromolecular scale
SYNTHESIS OF MACROMOLECULES: DNA, RNA AND PROTEIN

53. Characteristics of Genomes
- First sequenced genome (1995)
- Smallest free living organism

54. Features of the E. coli Genome
rRNA & tRNA

55. Features of the Human Genome
Based on NCBI assembly Build 36 (released 2005) (

56. WHAT ARE TYPICAL PROTEIN CONCENTRATIONS?

57. Protein Concentration in E. coli
Cells represent a fairly dense solutions of proteins
Concentration of total protein in cells falls in the range: 200 – 400 mg/ml
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

58. Size distribution of ORF or Protein sizes in E. coli

59. 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.

60. Publicly Available Proteomic Resources

61. What are Typical Nucleic Acid Concentrations?

62. References: PMID ;

63. WHAT ARE TYPICAL SYNTHETIC FLUXES OF MACROMOLECULES?

64. Typical Fluxes: DNA synthesis
The E. coli genome can be replicated in 40 min with 2 replication forks – the rate of DNA polymerase is:
The rate of RNA polymerase is much slower:

65. Protein Synthesis in E. coli
The rate of the ribosome is on the order of peptide bonds/ribosome/sec in rapidly growing E. coli.
The amount of ribosomes present in E. coli depends vastly on the growth rate: on the order of: 7x103 – 7x104 ribosomes/cell
The total amount of peptide bonds that are formed in E. coli as a function of growth rate can be estimated:
This value is equivalent to:

66. Protein Synthesis in Mammalian cell
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
The rate of the ribosome is 20 peptide bonds/cell/sec
The ribosomal spacing is nucleotides/mRNA
This leads to the protein production rate in murine cell:

67. What are Typical Turnover Times?

68. Mean mRNA half-life by gene functional class in E. coli
Adopted from: Bernstein et al. 2002, PNAS

69. CELL GROWTH AND PHENOTYPIC FUNCTIONS
The whole-cell scale
CELL GROWTH AND PHENOTYPIC FUNCTIONS

70. Phenotypic characteristics of E
Phenotypic characteristics of E. coli: Aerobic (60 min) and Anaerobic profile (90 min)

71. 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:
DNA:
Stable RNA
mRNA
chromosome
for a grand total of approximately 9.26 x 107 nucleotides/cell for synthesis of RNA and DNA molecules for one cell.

72. 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:

73. Cell doubling time: 60 minH+
-230 mV -> 105 V/cm
Energy production: 0.2 – 1.0 pW/µm3
ATP Production rate:
x 1010 molecules/cell/h
METABOLISM
Glycolytic flux: 3 x 109 molecules/cell/h
ADP
ATP H+
DNA replication rate:
900 bp/sec/fork
Nucleotide Flux:
5 x 108 nucleotides/cell/h
Protein
Amino Acid Flux:
9 x 108 amino acid/cell/h
DNA
Protein production rate:
3 x 106 proteins/cell/h
tRNA
Fraction of RNAP synthesizing tRNA/rRNA:
RNA Polymerase rate:
5 x 108 nucleotides/cell/h
Cell doubling time:
60 min
mRNA
Ribosome rate: 3 x 109 peptide bonds/cell/h

74. Overall metabolic rates in E. coli:
Implications for bioprocessing
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

75. 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.

76. FROM BACTERIA TO MAMMALS

77. Metabolic rate of major organs

78. Size range of living organisms
Figure taken: K. Schmidt-Nielsen, “Why is animal size so important”, 1984

79. Metabolic rate and body size
Figure taken: K. Schmidt-Nielsen, “Why is animal size so important”, 1984

80. 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.

81. O-OF-MAGNITUDE: some examples

82. 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.

83. 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...

84. And $1 BILLION dollars... now we're really getting somewhere...

85. Ladies and gentlemen... I give you $1 trillion dollars...
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.