Tsvi Tlusty, Physical Biology Gidi Lasovski
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Transcript of Tsvi Tlusty, Physical Biology Gidi Lasovski
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A simple model for the evolution of
molecular codes driven by the interplay
of accuracy, diversity and costTsvi 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 Biology1. A signaling protein binds to a gene
2. The RNA polymerase generates mRNA from the gene
3. The mRNA exits the nucleus of the cell
4. 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
Non PolarPolarBasicAcidic
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The genetic code - similarity
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The fitness of molecular codesThree 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 hereWe want to make sure the errors would be less
likely where they’re 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 -> 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 - I’m hungry
A medium lack of sugar - I’m starving
A large lack of sugar – Let’s 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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wc is equivalent to the temperature
J/wc = 1 is the phase transition: “liquid” (the non coding state) J/wc < 1
“solid” (the coding state) J/wc > 1
Ψ – the order parameter
H – the fitness
C – the cost
D – the diversity
L – the error load
The result is an Ising like model
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Possible experiment Take a bacteria with the
transcription factor i. Duplicate the gene that codes i,
let’s 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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Using Lagrange multipliers:
H = −L + WD · D − WC · CC is the reduction of entropy, so WC is equivalent to the
temperature (WCC ~ TdS)
Diversity
D = Σi,j,α,β(1 − δij )piαpjβcαβ
Error loadL = Σi,j,α,β rijpiαpjβcαβ
Cost
C = Σiα piα ln(piα/pα)
Eiα ln ∼ piα
pα = ns-1 Σj pjα
rij – the probability to read i as j
Piα – the probability for i to be mapped to α is
Cαβ – the cost of misinterpreting α as β
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Additional slides for the mathematical model
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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 < 1 “solid” (the coding state) J/wc > 1
Ψ – the order parameter
H – the fitness
C – the cost
D – the diversity
L – the error load
ψ = tanh (∗ J/wC · ψ )∗
H = c·J·ψ2 − wC[(1 + ψ) ln(1 + ψ)+ (1 − ψ) ln(1 − ψ)]
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Quantifying the code
Ns symbols (i, j, k..) mapped to Nm meanings (α, β..)
Piα - The probability for i to be mapped to α
ΣαPiα =1
In the non coding state, the prob. is constant 1/Nm
rij – the probability to read i as j.
Cαβ – the cost of misinterpreting α as β The total error load:
L = Σi,j,α,β rijpiαpjβcαβ
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/∂piα = 0 p∗
iα = z-1 p∗α exp(−Giα/wC) z = Σβ p∗
βexp(−Giβ/wC) Giα = 2Σj,β (rij − wD(1 − δij))pjβcαβ c = 0 c
c 0 r = 1−r r
r 1−r p = 0.5 1 + ψ 1 − ψ
1 − ψ 1 + ψ
ψ∗ = tanh (J/wC · ψ∗) J = c (1−2r + wD) wC∗ = J = (1 − 2r + wD) c
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General criteria
Qiαjβ =−(∂2H/∂piα∂pjβ) stops being positive definite
wC∗ = 2*nm-1 (λr
∗ + wD)|λc∗ |
λr∗ is the 2nd-largest eigenvalue of r
λc ∗ is the smallest eigenvalue of c - corresponds to the longest
wavelength – smallest error load