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Lecture #7

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  1. Lecture #7 Estimation and Orders of Magnitude

  2. Estimation

  3. Orders of Magnitude • Powers of 10: http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/ • Cell size and scale:http://learn.genetics.utah.edu/content/begin/cells/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? • What are Typical Turnover Times? • 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 (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-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 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:

  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 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?)

  15. Resource: BioNumbers database Source: http://bionumbers.hms.harvard.edu/ BioNumbers is coordinated and developed by Ron Milo at the Weizmann Institute in Israel.

  16. Orders of Magnitude SOME OVERALL OBSERVATIONS

  17. The Interior of a Cell:a crowded place Courtesy of David Goodsell http://mgl.scripps.edu/people/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. Small molecule scale METABOLISM

  24. The compounds WHAT ARE TYPICAL METABOLITE CONCENTRATIONS?

  25. 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!

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

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

  28. Size Distribution of Metabolites

  29. Publicly Available Metabolic Resources

  30. Reaction rates WHAT ARE TYPICAL METABOLIC FLUXES?

  31. What are Typical Turnover Times?

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

  33. 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

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

  35. 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

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

  37. 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

  38. Magnitude of the potential gradient • As presented above the potential is on the order of -220-240 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).

  39. ESTIMATING THE NUMERICAL VALUE OF KINETIC CONSTANTS

  40. Kinetic Constants of E. coli Enzymes • 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) 32 mM http://www.brenda-enzymes.info/

  41. Typical Enzyme Turnover Times 1 min ‘fast’ http://www.brenda-enzymes.info/

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

  43. WHAT ARE TYPICAL POWER DENSITIES?

  44. 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 • 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

  45. 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 • 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

  46. 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 m3per 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

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

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

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

  50. Features of the Human Genome Based on NCBI assembly Build 36 (released 2005) (http://www.ensembl.org/Homo_sapiens/index.html)