Stochasticity in Gene Expression

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Stochasticity in Gene Expression

• Arjun Raj

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Why consider stochasticity?

• Intracellular particle numbers are often quite
  low
• Genetic copy number and number of DNA
  transcription factor binding sites is very low.
• RNA levels are also often quite low (lt100
  molecules per cell.
• Consequences
• Kinetic mass action equations may no longer be
  accurate, since they only really give information
  about the mean.
• A more correct model may be a chemical master
  equation.

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Master Equation

• In practice, the master equation is not solvable
  in closed form for even moderately complicated
  systems.
• One could possibly solve the resultant system of
  ODEs numerically
• Unfortunately, the number of equations increases
  exponentially with the number of species.
• Often, the only solution left is to compute a
  large number of sample paths and then measure the
  statistics.
• Gillespies algorithm for generating sample paths
  is efficient and, amazingly, generates exact
  paths.

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Modifications of Gillespies algorithm

• By cleverly updating the propensities, one can
  reduce computation time of the first reaction
  method so that it is faster than the direct
  method.
• Should the particle numbers be large enough that
  the propensities change by a relatively small
  amount, one can leap ahead in the time history of
  the system by assuming propensities are constant
  (?-leap method).
• Should the particle numbers be even larger, one
  can replace the master equation with a
  Langevin-type equation.

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Measurements of noise

• Phenotypic variation is commonly observed in
  clonal populations.
• Question how much of the noise is due to
  transcriptional noise and how much is due to
  translational noise?

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Measurements in Bacillus subtilis

• Experiments introduced GFP chromosomallly into B.
  subtilis under the control of a IPTG inducible
  promoter (Pspac), allowing for control of
  transcriptional efficiency.
• Translational efficiency varied by altering RBS.

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Results

• Noise quantitated using Fano factor of
  ltp2gt/ltpgt.
• Measures deviation from Poisson behavior.
• Noise seems largely independent of
  transcriptional efficiency, but shows a linear
  correlation with translational efficiency.

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Why linear behavior?

• Can solve the chemical master equation using
  moment generating functions.
• Results in ltrgt being kR/?R, obeying a Poisson
  distribution.
• However, p obeys a different distribution

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Situation appears to be different in eukaryotes

• Contructed a more complicated genetic control
  mechanism in yeast.
• Transcription mechanisms are different in
  eukaryotes, so might have different noise
  characteristics.

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Results

• Measured noise using Fano factor as a function of
  transcriptional efficiency by varying amount of
  galactose and Atc in growth medium.
• Also varied translational efficiency by using
  different codon variants of yEGFP gene.

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Simulations

• Used a model involving preinitiation complexes.
• Were able to reproduce data (with appropriate
  parameters).

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Question how much is the stochasticity is really
due to the inherent randomness of transcription
and translation?

• Randomness in gene expression may be due to
  extrinsic factors which arent related to the
  actual stochastic nature of genetic expression
• Amount of RNAP, ribosomes
• State of cell in cell cycle
• State of degradation machinery
• How can one distinguish this from the intrinsic
  noise due to the randomness in the fundamental
  chemistry of gene expression?

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Results in E. coli

• Inserted YFP and CFP into E. coli genome in
  identical way.
• Can measure contributions of intrinsic and
  extrinsic noise by measuring relative CFP and YFP
  levels within a single cell.

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Results
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Results

• Shows a decrease in intrinsic noise as
  transcriptional efficiency increases.
• Hard to compare to other studies, since noise is
  measured differently.

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So what?

• Noise has been shown to be important in switching
  behavior of ?-phage life cycle in E. coli.
• Noise must be controlled by cells to produce
  reliable behavior
• Circadian rhythms
• Development

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Repressilator