Tsvi Tlusty, Physical Biology Gidi Lasovski

1. A simple model for the evolution ofmolecular codes driven by the interplayof 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 mRNAand 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: • RNAPolymerase • 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

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 they’re 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 -> 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 - I’m hungry A medium lack of sugar - I’m starving A large lack of sugar – Let’s 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 Ψ – 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 < 1 • “solid” (the coding state) J/wc > 1

18. 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

19. 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 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 = c·J·ψ2 − wC[(1 + ψ) ln(1 + ψ)+ (1 − ψ) ln(1 − ψ)] Ψ – the order parameter H – the fitness C – the cost D – the diversity L – the error load ψ∗ = tanh (J/wC· ψ∗) 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

22. 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

23. Toy model (1 bit) P∗ - the optimal code, can be found by the derivation ∂HT/∂piα= 0 • p∗iα= z-1p∗αexp(−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 = 0.51 + ψ 1 − ψ 1 − ψ 1 + ψ • ψ∗ = tanh (J/wC · ψ∗) • J= c (1−2r + wD) • wC∗= J = (1 − 2r + wD) c

24. 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