Lecture 7 Estimation and Orders of Magnitude

Lecture #7 Estimation and Orders of Magnitude

Estimation

Orders of Magnitude • Powers of 10: http: //micro. magnet. fsu. edu/primer/java/s cienceopticsu/powersof 10/ • Cell size and scale: http: //learn. genetics. utah. edu/content/begi n/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 approximately 5, 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 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 m. RNA in E. coli? • How many ATP synthase complexes are required for optimal growth on glucose in E. coli?

proteins/m. RNA: 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 m. RNA: 0. 03 – 0. 05 – Source: PMID 11713332 • Total m. RNA nucleotide residues: 2, 190, 000 – 3, 650, 000 nt • Average length of m. RNA: 1, 100 nt • Number of m. RNA / cell: 2000 -3300 • Average length of protein: 367 AA • Number of proteins / cell: 2. 4 million • 725 -1200 proteins / m. RNA:

proteins/m. RNA: method 2 • Average length of m. RNA: 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 m. RNA: 6 minutes (360 seconds) • Mean lifetime of m. RNA = 519 seconds (half-life / ln 2) • 519 proteins/m. RNA

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/g. Dwh – Input: Aerobic + 10 mmol glucose / g. Dwh • With 2. 8*10 -13 g. Dw/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: Bio. Numbers database Species # Bio. Numbers E. coli 920 H. sapiens 667 S. cerevisiae 394 Source: http: //bionumbers. hms. harvard. edu/ Bio. Numbers 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 -2µm 3 Glucose turnover in rapidly growing E. coli: • Extracellular Glucose concentration: 1 -5 m. M (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 (2 ATP+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 m. V • Bi-lipid layers become physically unstable at 280 m. V

Magnitude of the potential gradient 1. As presented above the potential is on the order of -220 -240 m. V across the energy transducing membrane. 2. The thickness of the lipid bi-layer is on the order of 7 nm. 3. So the potential gradient across this membrane is: • 230 m. V/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 m. M • 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 ‘fast’ http: //www. brenda-enzymes. info/

The Distributions of Gibbs Free Energies in i. AF 1260 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 μm 3 • The energy of the phosphate bond is about 52 k. J/mol ATP 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 μm 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 μm 3 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 m 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 m. M • Maximal fluxes are about a million molecules per m 3 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 p. W/m 3

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 r. RNA & t. RNA

Features of the Human Genome 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 k. Da. • 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 K 12 MG 1655. 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:

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: 7 x 103 – 7 x 104 ribosomes/cell 3. The total amount of peptide bonds that are formed in E. coli as a function of growth rate can be estimated: 4. This value is equivalent to:

Protein Synthesis in Mammalian cell 1. The total amount of m. RNA from a single gene in the cytoplasm of the murine cell is on the order of 40, 000 m. RNAs/cell 2. The rate of the ribosome is 20 peptide bonds/cell/sec 3. The ribosomal spacing is 90 -100 nucleotides/m. RNA 4. This leads to the protein production rate in murine cell:

The whole-cell scale CELL GROWTH AND PHENOTYPIC FUNCTIONS

PHENOTYPIC CHARACTERISTICS OF E. COLI: AEROBIC (60 MIN) AND ANAEROBIC PROFILE (90 MIN)

Synthesis of an E. coli Cell: order-of-magnitude estimation of fluxes • • There are 3. 0 x 106 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 m. RNA 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 m. V -> 105 V/cm Energy production: 0. 2 – 1. 0 METABOLISM 0. 8 - 4 x 1010 molecules/cell/h H+ ATP Production rate: Glycolytic flux: 3 x 109 molecules/cell/h p. W/µm 3 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 t. RNA Protein production rate: 3 x 106 proteins/cell/h Fraction of RNAP synthesizing t. RNA/r. RNA: 0. 28 -0. 77 RNA Polymerase rate: 5 x 108 nucleotides/cell/h Cell doubling time: m. RNA Ribosome rate: 3 x 109 peptide bonds/cell/h 60 min

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

Limits on Volumetric Productivity • Anaerobically E. coli has substrate uptake rate (SUR) of: 15 – 20 mmol Glucose/ g. DW/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/µm 3, while enzymes have an average concentration of 2000 molecules/µm 3. • 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/µm 3 /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 $100 million is a little more little pile is $1 million dollars respectable. It fits neatly on (100 packets of $10, 000). 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. 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.