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Gene expression noise, regulation, and noise propagation - Erik van Nimwegen

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Gene expression noise, regulation, and noise propagation - Erik van Nimwegen

  1. 1. Gene  expression  noise,   regula0on,  and  noise  propaga0on   Erik  van  Nimwegen   Biozentrum,  University  of  Basel,   and  Swiss  Ins8tute  of  Bioinforma8cs   Basel   Our  group  
  2. 2. Cartoon  of  the  steps  in  gene  expression   Gene  X   RNA   polymerase   Gene  X   RNA   polymerase   mRNA  gene  X   Transcrip0on  rate:     rλ mRNA  decay  rate   Protein  X   Transla0on  rate   Protein  decay  rate:   pλ pµ rµ
  3. 3. Gene  expression  differen3al  equa3ons   •   P  =  Amount  of  protein  X.   •   R  =  Amount  of  mRNA  X.   •   P  increases  due  to  transla0on  of  mRNA  and  decreases  due  to  protein  decay.   p p dP R P dt λ µ= − •   R  increases  due  to  transcrip0on  and  decreases  due  to  mRNA  decay.   r r dR R dt λ µ= − Steady-­‐state:     P = λr λp µr µp R = λr µr •   In  reality  there  are  are  really  an  integer  number  p(t)  of  proteins  at  0me  t,  and  r(t)  mRNAs.   •   Numbers  may  be  small,  e.g.  there  is  only  one  copy  of  the  gene  in  the  DNA.   •   The  RNA  polymerases,  ribosomes,  and  mRNAs    are  tumbling  around  in  the  cell,        constantly  bumping  into  other  molecules  (i.e.  following  Brownian  mo0on).   Discreteness  and  Stochas3city:  
  4. 4. Surprise  surprise:     Gene  expression  is  stochas3c   Low copy Plasmid •  GFP  fluorescence  per  cell  propor0onal  to  protein   number.   •  Not  surprisingly,  fluctua0ons  are  observed  between   cells.   •  What  kind  of  fluctua0ons  would  one  expect  in  a   simplest  possible  model?      
  5. 5. Stochas3c  transcrip3on  and  decay   Gene  X   RNA   polymerase   Gene  X   RNA   polymerase   mRNA  gene  X   Probability            per  unit  0me  to  transcribe  a   new  mRNA.   Differen0al  equa0on  for  the  distribu0on:   1 1 ( ) ( ) ( 1) ( ) ( ) ( )n r n r n r r n dP t P t n P t n P t dt λ µ λ µ− += + + − + Probability  that  there  are  n  mRNAs  at  0me  t:   rλ rµ Pn (t) Probability          per  mRNA  per  unit   0me  that  it  will  decay.    
  6. 6. Steady-­‐state  is  Poisson  distribu3on   Probability  to  have  n  mRNAs:   Pn = 1 n! λr µr ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ n e −λr /µr Mean:   n = λ µ Variance:   var(n) = n = λ µ Standard-­‐devia3on:  σ (n) = n 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 Number of mRNA n Probability 0 2 4 6 8 10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Number of mRNA n Probability 0 5 10 15 20 25 30 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Number of mRNA n Probability λr µr = 0.1 10r r λ µ = λr µr =1
  7. 7. (Shahrezaei,  Swain  PNAS  2008)   Transla3on  amplifies  mRNA  fluctua3ons   mean  and  variance:   a = λr µp b = λp µp “burst  size”:  transla0ons  per  mRNA  life0me.   n = ab, var(n) = (b+1) n λr µr µp transcrip0on   mRNA  decay   transla0on   protein  decay   λp λr µr λp µp •  Proteins  are  oZen  long-­‐lived:  approxima0on  protein-­‐decay  slow  rela0ve  to  mRNA  decay.   •  Solu0on  in  terms  of  two  ra0os:   Transcrip0on  events  per  protein  life0me.   Pn = Γ(a + n) Γ(a)n! b b +1 ⎛ ⎝⎜ ⎞ ⎠⎟ n 1− b b +1 ⎛ ⎝⎜ ⎞ ⎠⎟ a noise:   η(n) = σ(n) n = var(n) n 2 = b +1 n
  8. 8. mRNAs  per  cell  for  E.  coli   hp://book.bionumbers.org/  
  9. 9. Typical  genes  have  less  than  1     mRNA  per  cell  in  E  coli   Fluorescently  labeling  single   mRNAs  (Fluorescence  In   Situ  Hybridiza0on).   Coun0ng  mRNAs  per  cell   under  the  microscope.   Mean  mRNAs  per  cell   Taniguchi  et  al,  Science  (2010)   From:  Milo  and  Phillips,  Cell  Biology  by  the  numbers    
  10. 10. Some  addi3onal  numbers  for  E.  coli   •   RNA  polymerases  per  cell:  1’500-­‐10’000  (depending  on  growth  rate).   •   Ribosomes  per  cell:  14’000  (1  doubling  per  hour)  –  45’000  (2  doublings  per  hour).   •   mRNA  decay  rate:  1-­‐15  minutes  half-­‐life.     •  Protein  decay  rate:  typically  a  few  hours.     •  Protein  dilu0on  rate:  cell  doubling  0me,  i.e.  30  min  to  2  hours.     Bernstein  et  al,  PNAS  (2002)   Taniguchi  et  al,  Science  (2002)   Distribu3on  mRNA  half-­‐lifes   Distribu3on  mean  proteins  per  cell  
  11. 11. Measuring  variability  within  and  across  cells   Two  3mes  the  same  promoter   Intrinsic  and  extrinsic  noise   •  Total  variance  in  fluorescence  per  cell  can  be  decomposed  into  two  parts:   •  Intrinsic  =  variance  within  cell:     •  Extrinsic  variance  =  the  rest,  i.e.  variability  across  cells:     vtot = var(g) + var(r) = vi + ve vi = 1 2 (g − r)2 ve = gr − g r Hey!  That  covariance  could  be  nega8ve!    How  can  a  variance  be  nega8ve?    
  12. 12. How  to  properly  infer     intrinsic  and  extrinsic  variance   Gives  orthodox  sta0s0cal  es0mators     that  can  give  nega0ve  es0mates.   A  Bayesian  solu3on  is  never  pathological  and  much  more  accurate  when  extrinsic  noise  is  small     Extrinsic:  Gaussian  distribu0on  of  mean  μi  across  cells  i:     Intrinsic:  Gaussian  devia0on  of  green  gi  and  red  ri  from  mean  μi:     P(gi ,ri | µi ) = 1 2πv exp − (gi − µi )2 + (ri − µi )2 2v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ Posterior  for  the  intrinsic  variance  v  and  extrinsic  variance  vμ:   P(v,vµ | D) = vµ + v / 2( ) −(n−1)/2 v−n/2 exp − n 4v (g − r)2 − n (2vµ + v) var r + g 2 ⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ Example  with  low  extrinsic  noise   Inference  based  on                              only.    (g − r)2 Bayesian  result.     Result  assuming  extrinsic  noise  known.    
  13. 13. Extrinsic  noise  implies     transcrip3on/transla3on/decay  rates    fluctuate   Extrinsic  noise  in  Elowitz  et  al:   Intrinsic  noise  falls  as  the  promoter  is  induced.   Extrinsic  noise  peaks  at  intermediate  induc0on.   R  Phillips  (Annu  Rev  Con  Mat  Phys,  2015)   •  Transcrip0on  rate  can  vary  when  the   promoter  switches  between  different  states.   •  Switching  rates  depend  on  concentra0ons  of   DNA  binding  proteins  (polymerases,  TFs).     •  These  concentra0ons  will  fluctuate  from  cell   to  cell.  
  14. 14. Noise  propaga3on   •  Regulatory  cascade:  Gene  1  induces  gene  2.  Gene  3  cons0tu0ve.   •  As  gene  1  is  induced,  its  own  noise  level  drops.   •  Gene  2  goes  through  an  intermediate  peak  in  noise  level.     •  Gene  3’s  noise  is  unaffected.       Interpreta3on:   At  intermediate  levels  of  gene  1,  the  promoter  of  gene  2  shows  most   switching  between  bound  and  unbound  states  and  most  sensi0vity  to   fluctua0ons  in  the  concentra0on  of  gene  1.  
  15. 15. Cells  are  not  sta3c:   Inves0ga0ng  stochas0c  regulatory  dynamics   Wish  list   •  Follow  growth  and  gene  expression  dynamics  in  single  cells  over  long  0me  scales.   •  Accurate  quan0fica0on.   •  Follow  different  cell  lineages  separately  to  allow  observa0on  of  rare  events.   •  Precise  dynamical  control  over  growth  environment.   Wang  et  al.  Robust  growth  of  Escherichia  coli.  Curr  Biol.  2010   The  mother  machine  
  16. 16. Our  extension:   The  Dual  Input  Mother  Machine  
  17. 17. Switching  growth  media  between     glucose  and  lactose   •  GFP/lacZ  fusion  reports  lac-­‐operon  expression.   •  Switch  glucose/lactose  every  4  hours.   •  Immediate  growth  arrest  at  first  switch  to  lactose.   •  Stochas0c  induc0on  of  lac-­‐operon  and  restart  of  growth.   •  Dilu0on  of  GFP/lacZ  during  glucose  phase.   •  No  more  growth  arrests  upon  later  switches.  
  18. 18. Automated  Image  Analysis:   The  Mother  Machine  Analyzer   Florian  Jug   Gene  Myers   MPI  Cell  Biology,  Dresden   •  Tracking  and  segmenta0on  done  in  parallel  using  a   single  objec0ve  func0on.   •  Interac3ve  cura3on:     •  User  input  interpreted  as  addi0onal  constraints.   •  Automa0c  re-­‐op0miza0on.    
  19. 19. Cells  expand  exponen3ally  during  their  cell  cycle   2 3 4 2 3 4 2 3 4 2 3 4 0 4 8 12 16 20 time (h) celllength(µm) 0.970 0.975 0.980 0.985 0.990 0.995 1.000 0.0 0.2 0.4 0.6 0.8 1.0 Pearson correlation exp. growth curve FractionCellCycles Cumula3ve  correla3on  coeff.   of  log(size)  vs  3me   Example  growth  dynamics  of  log-­‐size  vs  3me   Roughly  two-­‐fold  variability  in  growth  rates  
  20. 20. Fluorescence  roughly  tracks  cell  size   but  produc3on  fluctuates  significantly   Approximately  4-­‐fold  varia3on  in  produc3on  rate   Examples  of  total   fluorescence  against  0me   for  single  cells  growing  in   lactose.   Distribu0on  of  GFP  molecules  produced  per   second.  
  21. 21. Distribu3on  of  total  fluorescence     and  fluorescence  concentra3ons   5000 10000 15000 20000 25000 30000 35000 0.00000 0.00005 0.00010 0.00015 Fluorescence HAUL Probabilitydensity Total Fluorescence Distribution m=10'616, s=2911, sêm=0.274 8.5 9.0 9.5 10.0 10.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Log Fluorescence HAUL Probabilitydensity Total Log Fluorescence Distribution m=9.23,s2 =0.07 4000 6000 8000 10000 12000 0.0000 0.0002 0.0004 0.0006 0.0008 Fluorescence concentrationHAUêmicronL Probabilitydensity Fluorescence Concentration Distribution m=4278, s=661, sêm=0.154 8.0 8.5 9.0 9.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Log Fluorescence concentrationHAUêmicronL Probabilitydensity Log Fluorescence Concentration Distribution m=8.35,s2 =0.022 Very  roughly  log-­‐normal  distribu0ons.  Concentra0on  has  significantly  less  varia0on.  
  22. 22. Measuring  transcrip3on     from  all  E.  coli  promoters  in  single  cells   •  GFP  fluorescence  per  cell  propor0onal  to  protein  number.   •  GFP  levels  of  single  cells  can  be  measured  in  high-­‐throughput  using  FACS.   •  Quan0ta0vely  characterize  the  distribu0on  of  expression  levels  across  single  cells,  for   all  E.  coli  promoters.   ORF1   ORF2   ORF4   E. coli genomeORF3   Plasmid Zaslaver et al. 2006 Silander  et  al.  PLoS  genet  2012   Wolf  et  al.  eLife  2015  
  23. 23. FACS:   Measuring  and  selec3ng  single  cells   •  Cells  move  one-­‐by-­‐one  in  a  flow  channel.   •  Each  cell  passes  in  front  of  a  laser  and  its  fluorescence  is   measured.     •  By  selec0vely  charging  par0cles  based  on  their  measured   fluorescence,  one  can  select  cells  whose  fluorescence  lies  in   a  certain  range.  
  24. 24. Gene  expression  distribu3ons     for  two  example  promoters   µ1 µ2 σ1 σ2 Promoter 1 Promoter 2
  25. 25. Means  and  variances  of   na3ve  E.  coli  promoters   •  Variance  in  log-­‐expression  in  shows  a  trend  of  decreasing  with  mean  expression.   •  Different  promoters  with  same  mean  can  show  significantly  different  variance.   •  There  seems  to  be  a  clear  lower  bound  on  variance  as  a  func0on  of  mean.   5 6 7 8 9 10 11 0.0 0.2 0.4 0.6 0.8 Mean Log@GFP IntensityD VarianceLog@GFPpercellD background   2  *  background  
  26. 26. 7 8 9 10 11 12 13 0.0 0.2 0.4 0.6 0.8 Mean Log@proteins per cellD VarianceLog@proteinspercellD 7 8 0.0 0.2 0.4 0.6 0.8 Excessnoise Means  and  variances  of   na3ve  E.  coli  promoters   Red  curve:       σab 2 = 0.025, b = 450 n = ab, var(n) = (b+1) nAt  constant  transcrip0on/transla0on/decay  rates:     Assume  a  and  b  both  fluctuate:   var(n) = (b +1) n +σab 2 n 2 nmeas = nbg + n + ε var(n) var log(nmeas )⎡⎣ ⎤⎦ = σab 2 1− nbg nmeas ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 + (b +1) nmeas 1− nbg nmeas ⎛ ⎝ ⎜ ⎞ ⎠ ⎟
  27. 27. Noise  levels  vary  across   na3ve  E.  coli  promoters   7 8 9 10 11 12 13 0.0 0.2 0.4 0.6 0.8 Mean Log@ proteins per cellsD Excessnoise Excess  noise  (variance  –  lower  bound  as  func.  mean)   Selec3on  on  noise  levels   High  noise   DriZ?  Selected  for  noise?   Low  noise.   Selec0on  to  minimize  noise?   What  noise  would  one  get  without  selec3on?     Evolve  synthe8c  promoters  in  a  precisely  controlled  selec0ve  environment.  
  28. 28. Directed  evolu0on  of  promoters     that  express  at  a  desired  level   *        *    *   28  
  29. 29. Evolu0on  of  popula0on  expression  levels   Selec0ng  for     Medium  expression   29   Selec0ng  for   High  expression  
  30. 30. Expression  distribu0ons  of     individual  synthe0c  promoters   •  We  isolated  ~400  clones  from  evolu0onary  runs  for  both  medium  and  high  expression.   •  Measured  each  clone’s  expression  distribu0on.     How  do  noise  levels  of  synthe3c  promoters  compare  with  those  of  na3ve  promoters?    
  31. 31. Na0ve  promoters   Synthe0c  promoters   •  Synthe0c  promoters  were  not  selected  on  their  noise  proper0es.   •  Low  noise  is  the  default  behavior  of  E.  coli  promoters.   •  Selec0on  must  have  acted  so  as  to  increase  the  noise  levels  of  some  na0ve  promoters.   Iden0cal  distribu0ons  at  the     low  noise  end.   High  noise  enriched  in     na0ve  promoters.     Selec0on  caused  increased  noise  in     a  substan0al  frac0on  na0ve  promoters   What  is  `special’  about  na3ve  promoters  that  show  high  noise?  
  32. 32. Noisy  genes  have  more  regulatory  inputs   •  185  E.  coli  transcrip0on  factors  (TFs).   •   4123  known  regulatory  interac0ons  TF  →  promoter.   Genes  with  higher  noise  have  (on  average)  higher  numbers  of  known  regulatory  inputs.   2  or  more  inputs   1  known  input   no  known  inputs   synthe0c  proms.   Why  is  there  a  general  associa3on  between  noise  and  regula3on?   Why  did  selec3on  cause  noise  to  increase?  
  33. 33. Noise-­‐propaga3on:  nuisance  or  opportunity?     Noise  as  an  unavoidable  side-­‐effect  of  regula3on   •  Explains  the  general  associa0on  of  noise  and  regula0on.   •  `Fluctua0on-­‐dissipa0on  rela0on’:  Genes  that  need  complex  regula0on  unavoidably  couple   to  the  noise  in  their  regulators.   •  Generally  assumed  to  be  detrimental:  reduces  the  accuracy  of  regula0on.   Stochas3city  as  a  bet-­‐hedging  strategy   •  Phenotypic  diversity  can  generally  be  selected  for  in  fluctua0ng  environments.   •  Maybe  noise-­‐propaga0on  can  be  beneficial  in  some  circumstances?   Let’s  do  some  theory  on  how  gene  expression  noise  affects  fitness  
  34. 34. Fitness  func0on   in  a  single  environment   f (x | µ*,τ ) = exp − (x −µ* )2 2τ 2 " # $ % & ' p(x | µ,σ ) = 1 2πσ exp − (x −µ)2 2σ 2 " # $ % & ' f (µ,σ | µ*,τ ) = dxp(x | µ,σ ) f (x | µ*,τ ) =∫ τ 2 τ 2 +σ 2 exp − (µ −µ* )2 2(τ 2 +σ 2 ) # $ % & ' ( The  fitness  of  a  promoter  `genotype’  (frac0on  of  its  cells  selected)  is  a  convolu0on  of  these   two  func0ons  (approx.  area  on  the  intersec0on):   Fitness  (probability  to  be  selected):   Promoter  expression  distribu0on:   σ = 0.1 µ µ* τ
  35. 35. Moving  the  mean  toward    the    desired  level  always  increases  fitness   f (µ,σ | µ*,τ ) = τ 2 τ 2 +σ 2 exp − (µ −µ* )2 2(τ 2 +σ 2 ) " # $ % & ' 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 0.0 0.2 0.4 0.6 0.8 1.0 Log expression ExpressionêSelectionprobability 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 0.0 0.2 0.4 0.6 0.8 1.0 Log expression ExpressionêSelectionprobability f (µ = 8.0,σ = 0.1) = 0.066 f (µ = 8.1,σ = 0.1) = 0.174
  36. 36. 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 0.0 0.2 0.4 0.6 0.8 1.0 Log expression ExpressionêSelectionprobability 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 0.0 0.2 0.4 0.6 0.8 1.0 Log expression ExpressionêSelectionprobability At  op0mal  mean     minimal  noise  is  preferred   f (µ,σ | µ*,τ ) = τ 2 τ 2 +σ 2 exp − (µ −µ* )2 2(τ 2 +σ 2 ) " # $ % & ' f (µ = 8.15,σ = 0.1) = 0.196 f (µ = 8.15,σ = 0.025) = 0.625
  37. 37. As  mean  moves  away  from  the  op0mum   there  is  a  bifurca0on  to  nonzero  op0mal  noise   f (µ,σ | µ*,τ ) = τ 2 τ 2 +σ 2 exp − (µ −µ* )2 2(τ 2 +σ 2 ) " # $ % & ' f (µ = 8.0,σ = 0.05) = 0.0077 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 0.0 0.2 0.4 0.6 0.8 1.0 Log expression ExpressionêSelectionprobability f (µ = 8.0,σ = 0.1) = 0.066 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 0.0 0.2 0.4 0.6 0.8 1.0 Log expression ExpressionêSelectionprobability `Bifurca3on’  in  op3mal  σ     When                                              ,  the  op0mal  noise  level  is   non-­‐zero:   0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Expression deviation »mu-mu*» Optimalsigma Op3mal  σ       σ* = (µ −µ* )2 −τ 2 τ = 0.05 τ = 0.2µ −µ* ≥ τ
  38. 38. Variable  environment:   Fitness  of  an  unregulated  gene   log f (µ,σ )[ ]= − (µ −µe )2 2(τ 2 +σ 2 ) + 1 2 log τ 2 τ 2 +σ 2 " # $ % & 'Log-­‐fitness  in  a  variable  environment:   Assuming  no  regula0on,  op0mal  mean  equals   Log-­‐fitness  becomes:         Op3mal  noise  matches  the  varia3on  in  desired  expression  levels:     log f (µ,σ )[ ]= − var(µe ) 2(τ 2 +σ 2 ) + 1 2 log τ 2 τ 2 +σ 2 " # $ % & ' This  is  the  bet  hedging  scenario.  But:   Wouldn’t  it  be  beer  to  evolve  gene  regula0on?     σopt 2 = var(µe )−τ 2 µ = µe
  39. 39. Effects  of  coupling  a  gene  to  a  regulator   Regulator’s  ac0vity   Gene  coupled  to  the  regulator.   Gene  without  regula0on   TF   TF Two  main  effects  on  the  gene’s  expression:   1.  Condi3on-­‐response:  Mean  depends  on  regulator’s  (condi0on-­‐dependent)  ac0vity.   2.  Noise-­‐propaga3on:  Noise  increases  due  to  propaga0on  of  the  regulator’s  noise.   We  developed  a  general  theory  to  calculate  how  these  effects  conspire  to  affect  fitness.    
  40. 40. Fitness  depends  on  only  4  effec3ve  parameters   Varia0on  in  desired  levels:  V     στ 1.  Expression  mismatch:   Y 2 = V σ 2 +τ 2 Varia0on  in  regulator  levels:  Vr     σr 2.  Signal-­‐to-­‐noise  of  the  regulator:   S2 = Vr σr 2 3.  Correla3on  regulator/desired  levels:  R Fitness  effect  of  the  regulatory  interac3on:   4.  Coupling  strength:   X log[ f ]= − 1 2 Y 2 (1− R2 )+ SX − RY( ) 2 (1+ X 2 ) − 1 2 log 1+ X 2" # $ % Scenario:  Start  with  unregulated  promoter.  What  fitness  can  be  obtained  by  coupling  to   regulator  with  signal-­‐to-­‐noise  S  and  correla0on  R?  
  41. 41. Fitness  with  op0mal  coupling  to  a  regulator   of  given  correla0on  R  and  signal-­‐to-­‐noise  S   Fitness  of  the   unregulated  promoter.   Y=4   Perfect    correla0on   No   correla0on   Noisy     regulator   Precise     regulator  
  42. 42. Coupling  to  a  near  op3mal  regulator:   condi3on-­‐response  effect   Y=4   TF   TF σtot = 0.16 R = 0.95 S = 3.3 Fitness  of  the   unregulated  promoter.  
  43. 43. Coupling  to  a  noisy  uncorrelated  regulator:   noise-­‐propaga3on  implements  bet  hedging  strategy   Y=4   TF   TF σtot = 0.55 R = 0 S = 0.19 Fitness  of  the   unregulated  promoter.  
  44. 44. Intermediate  case:   a  moderately  correlated  regulator   Y=4   TF   TF σtot = 0.23 R = 0.64 S = 2.45 Fitness  of  the   unregulated  promoter.  
  45. 45. Op0mal  S  at  a  given  R.   Y=4   Condi3on-­‐response  and  noise-­‐propaga3on     typically  act  in  concert   Regulator     too  noisy.   Regulator  not   noisy  enough.   •  Noise-­‐propaga0on  is  oZen  func8onal,  ac0ng  as  a  rudimentary  form  of  regula0on.   •  De  novo  evolu0on  of  regula0on:  Star0ng  from  pure  noise-­‐propaga0on  (R=0,S=0)   there  is  a  con0nuum  of  solu0ons  of  increasing  accuracy  along  which  condi0on-­‐ response  and  noise-­‐propaga0on  op0mally  complement  each  other.         •  Regulated  genes  are  noisy  because,  whenever  the  condi0on-­‐response  is  imperfect,   maximal  fitness  requires  noisy  regulators.   Summary  Theory:  
  46. 46. 0 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 Y: Expression mismatch R:Correlationofregulator'sexpressionwithdesired-levels σtot 2 =σ 2 Low  noise  regime:   Promoters  with  low  expression  mismatch  Y<1  `do  not  bother’  to  be  regulated.   For  extremely    correlated  regulators,  zero  noise-­‐propaga0on  is  the  op0mum.   Phase  diagram  of  final  noise   aZer  coupling  to  regulators  with  op0mal  noise  levels.  
  47. 47. 0 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 Y: Expression mismatch R:Correlationofregulator'sexpressionwithdesired-levels σtot 2 =σ 2 Noise-­‐propaga3on  regime:   The  final  noise  level  matches  the  frac0on  of  variance  in  desired  levels  not  tracked  by  the   condi0on-­‐response.   σtot 2 = (1− R2 )var(µe )−τ 2 Phase  diagram  of  final  noise   aZer  coupling  to  regulators  with  op0mal  noise  levels.   Amount  of  regula3on  required.   Variance  in  desired  levels   Selec3on  tolerance   Limited  accuracy  of  the  condi3on-­‐response.   Frac3on  variance  not  tracked  by  regula3on.  
  48. 48. Conclusions   signal   regulator   •  We  evolved  synthe0c  promoters  de  novo  in  E.  coli  under  carefully-­‐ controlled  selec0ve  condi0ons.   •  No  evidence  E.  coli  promoters  have  been  selected  to  lower  noise.     •  Regulated  genes  have  been  selected  to  increase  noise.     Experimental  observa3ons   Theory   •  Coupling  a  regulator  to  a  target  promoter  has  two  effects:   1.  Condi0on-­‐response.   2.  Noise-­‐propaga0on.   •  Noise-­‐propaga0on  alone  can  act  as  a  rudimentary  form  of  regula0on.   •  Accurate  regula0on  can  evolve  smoothly  along  a  con0nuum  in  which   noise-­‐propaga0on  and  condi0on-­‐response  act  in  concert.     •  Whenever  the  condi0on-­‐response  has  limited  accuracy,  noisy   regula0on  is  preferred.   •  Explains  the  general  associa0on  between  noise  and  regula0on.    
  49. 49. Thank  you!   Luise  Wolf                Olin  Silander   Theory/computa3on  PhD  and  post-­‐doc  posi3ons  available!   This  work:   Our  group  

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