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PPT – A simple model for the evolution of molecular codes driven by the interplay of accuracy, diversity and cost PowerPoint presentation | free to download - id: 715cc8-NzA2Z

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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

The main idea

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

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

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

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?

The genetic code

Acidic Basic Polar Non Polar

The genetic code - similarity

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

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

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

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!

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

Cost

- Car insurance the cost of improving the

robustness of your driving - Another example is the price of ink and space in

my demonstration

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

Summary

- The code has to be optimized at an equilibrium of

error load, diversity and cost.

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)

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

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

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)

Additional slides for the mathematical model

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

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

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

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