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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 of molecular codes driven by the interplay of accuracy, diversity and cost Tsvi Tlusty, Physical Biology Gidi Lasovski – PowerPoint PPT presentation

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


1
A simple model for the evolution of molecular
codes driven by the interplay of accuracy,
diversity and cost
  • Tsvi Tlusty, Physical Biology
  • Gidi Lasovski

2
The main idea
  • Understanding molecular codes
  • Their evolution and the forces that affect them

3
  • What is a molecular code
  • The genetic code
  • The fitness of molecular codes
  • The evolution and emergence of molecular codes
  • Suggested experimental verification

4
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

5
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?

6
The genetic code
Acidic Basic Polar Non Polar
7
The genetic code - similarity
8
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

9
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

10
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

11
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!

12
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

13
Cost
  • Car insurance the cost of improving the
    robustness of your driving
  • Another example is the price of ink and space in
    my demonstration

14
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

15
Summary
  • The code has to be optimized at an equilibrium of
    error load, diversity and cost.

16
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)

17
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

18
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

19
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)

20
Additional slides for the mathematical model
21
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

22
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

23
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

24
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
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