A simple model for the evolution of molecular codes driven by the interplay of accuracy, diversity and cost

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A simple model for the evolution ofmolecular
codes driven by the interplayof accuracy,
diversity and cost

• Tsvi Tlusty, Physical Biology
• Gidi Lasovski

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The main idea

• Understanding molecular codes
• Their evolution and the forces that affect them

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• What is a molecular code
• The genetic code
• The fitness of molecular codes
• The evolution and emergence of molecular codes
• Suggested experimental verification

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The Central Dogma of Molecular Biology

• A signaling protein binds to a gene
• The RNA polymerase generates mRNA from the gene
• The mRNA exits the nucleus of the cell
• A Ribosome reads the mRNA and creates a protein,
  with the help of tRNAs
• The tRNAs provide the Ribosome with amino acids,
  the building blocks of the protein

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What is a molecular code?

• The Genetic Code is a molecular code
• The symbols are A, U, C G
• The Machine
• RNA Polymerase
• Signaling molecules (proteins)
• mRNA
• Ribosome
• The output
• Proteins
• The cost of operation of the machine is the ATP
  and the tRNAs.
• The symbols encode Amino Acids redundantly
• 64 options only 20 amino acids
• for robustness reasons?

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The genetic code
Acidic Basic Polar Non Polar
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The genetic code - similarity
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The fitness of molecular codes

• Three parameters
• Error load
• Diversity
• Cost
• We define the fitness of the code as the linear
  combination of these three conflicting needs

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Error load

• When reading a number, we can misread 3 for 8 (or
  vice versa) anywhere
• 3838383838383838383838
• here or here
• We want to make sure the errors would be less
  likely where theyre more important
• 3838383838383838383838

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Error load

• Similar meaning should go with a similar (close)
  symbol, so that a small reading error would cause
  only a small understanding error.
• If this -gt signifies the deviation of sugar,
  which code would you prefer
• A or B

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Diversity

• Enables efficient and accurate delivery of
  different messages.
• A small lack of sugar - Im hungry
• A medium lack of sugar - Im starving
• A large lack of sugar Lets go to San Martin
• NOW!

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Diversity

• Enables the code to transmit as many different
  symbols as possible, equivalent to different
  symbols in a UTM
• Many different symbols less states of the
  machine
• More symbols also enable faster, more accurate
  control

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Cost

• Car insurance the cost of improving the
  robustness of your driving
• Another example is the price of ink and space in
  my demonstration

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Cost

• Strong binding takes up more energy to create and
  read
• The energy is proportional to the length of the
  binding site.
• The binding probability scales like e-E/T, E
  ln(p)
• Notice that diversity has its costs as well, more
  symbols means longer molecules

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Summary

• The code has to be optimized at an equilibrium of
  error load, diversity and cost.

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Quantifying the code

• Using Lagrange multipliers
• H -Load WD Diversity - WC Cost
• C is the reduction of entropy, so WC is
  equivalent to the temperature (WCC TdS)

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The result is an Ising like model
? the order parameter H the fitness C the
cost D the diversity L the error load

• wc is equivalent to the temperature
• J/wc 1 is the phase transition
• liquid (the non coding state) J/wc lt 1
• solid (the coding state) J/wc gt 1

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Possible experiment

• Take a bacteria with the transcription factor i.
• Duplicate the gene that codes i, lets call the
  duplicate j
• i, j control the response to A(t)
• If A(t) fluctuates strongly, i, j may evolve to 2
  different meanings - better control
• If A(t) fluctuates weakly, maybe one of them
  would be deleted.
• Experiment around the critical point

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rij the probability to read i as j Pia the
probability for i to be mapped to a is Caß the
cost of misinterpreting a as ß
Cost C Sia pia ln(pia/pa) Eialn pia pa ns-1 Sj pja Diversity D Si,j,a,ß(1 - dij )piapjßcaß Error load L Si,j,a,ß rijpiapjßcaß

• Using Lagrange multipliers
• H -L WD D - WC C
• C is the reduction of entropy, so WC is
  equivalent to the temperature (WCC TdS)

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Additional slides for the mathematical model
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H cJ?2 - wC(1 ?) ln(1 ?) (1 - ?) ln(1 -
?)
? the order parameter H the fitness C the
cost D the diversity L the error load ?
tanh (J/wC ?)

• J c (1-2r wD)
• wc is equivalent to the temperature
• J/wc 1 is the phase transition
• liquid (the non coding state) J/wc lt 1
• solid (the coding state) J/wc gt 1

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Quantifying the code

• Ns symbols (i, j, k..) mapped to Nm meanings (a,
  ß..)
• Pia - The probability for i to be mapped to a
• SaPia 1
• In the non coding state, the prob. is constant
  1/Nm
• rij the probability to read i as j.
• Caß the cost of misinterpreting a as ß
• The total error load
• L Si,j,a,ß rijpiapjßcaß
• Just like a ferromagnet r interaction, c
  magnitude p the spin
• Also prefers specific symbols L(rii) 0 only if
  i signifies a specific meaning

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Toy model (1 bit)

• P - the optimal code, can be found by the
  derivation ?HT/?pia 0
• pia z-1 pa exp(-Gia/wC) z Sß
  pßexp(-Giß/wC)
• Gia 2Sj,ß (rij - wD(1 - dij))pjßcaß
• c 0 c
• c 0
• r 1-r r
• r 1-r
• p 0.5 1 ? 1 - ?
• 1 - ? 1 ?
• ? tanh (J/wC ?)
• J c (1-2r wD)
• wC J (1 - 2r wD) c

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General criteria

• Qiajß -(?2H/?pia?pjß) stops being positive
  definite
• wC 2nm-1 (?r wD)?c
• ?r is the 2nd-largest eigenvalue of r
• ?c is the smallest eigenvalue of c - corresponds
  to the longest wavelength smallest error load