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Arvind Murugan Physics and the James Franck Ins3tute University of Chicago
Kine%c Proofreading Stochas%c algorithms Search strategies
Error correc'on through catastrophes Arvind Murugan Physics and - - PowerPoint PPT Presentation
Error correc'on through catastrophes Arvind Murugan Physics and - - PowerPoint PPT Presentation
Error correc'on through catastrophes Arvind Murugan Physics and the James Franck Ins3tute University of Chicago Kine%c Proofreading Stochas%c algorithms Search strategies Errors in enzyma'c reac'ons T G C T A - C - A - G - C - T - T -
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Errors in enzyma'c reac'ons
T - G - T - C - G - A - A - A - G - C A - C - A - G - C - T -
DNA polymerase
T G T C
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Errors in enzyma'c reac'ons
T - G - T - C - G - A - A - A - G - C A - C - A - G - C - T - T G C T
DNA polymerase
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Errors in enzyma'c reac'ons
T - G - T - C - G - A - A - A - G - C A - C - A - G - C - T - T G C T
DNA polymerase
E(A ≡ G) − E(A ≡ T) = ∆
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Errors in enzyma'c reac'ons
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Errors in enzyma'c reac'ons
T - G - T - C - G - A - A - A - G - C A - C - A - G - C - T - T G C T
DNA polymerase
E(A ≡ G) − E(A ≡ T) = ∆
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Errors in enzyma'c reac'ons
- Protein synthesis
- tRNA charging
- T-cell receptors
- ….
How do you reduce effec?ve error rate below (Fixed Δ)
e− ∆
kT
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E R1 E Ri ERi+1 ER Right product + E
(wrong substrate)W + E + R (right substrate)
EW EWi+1 EWi EW
1
Wrong product + E
!Δ
Energy difference Delta
Kine'c Proofreading
John Hopfield (1974) Jacques Ninio (1975)
+ Right Substrate R Enzyme E + Wrong Substrate W Right Product Wrong Product
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Kine'c Proofreading
John Hopfield (1974) Jacques Ninio (1975)
η ∼ e−2∆
f
EW EWi+1 EWi EW1
(wrong substrate, eg. ‘G’)W + E + R (right substrate, eg. ‘T’)
ER1 ERi ERi+1 ER Right product + E Wrong product + E
f
e∆ e∆
+ Right Substrate R Enzyme E + Wrong Substrate W Right Product Wrong Product
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General Principle
A.M, D.Huse, S.Leibler, PNAS 2012
E ER EW
W-Product + E R-Product + E
W R
ER2 ER1 ERi EW1 EW2 EWi EWj ERj
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E ER EW e∆ e
∆
W-Product + E R-Product + E
W R
ER2 ER1 ERi EW1 EW2 EWi EWj ERj
e
- ∆
e∆ e
∆
e
∆
e
∆
General Principle
A.M, D.Huse, S.Leibler, PNAS 2012
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General network
Red lines –checkpoints, in parallel. Start Finish e∆ e∆ e∆ e∆ e∆ e∆ e∆
Can n parallel paths, each with a failure rate η ∼ e−∆ be combined to give η ∼ e−n∆
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E+S ES
Error correc'ng kine'c limit
Reac?on is completed
- nly along green path.
Reac?on interrupted at red checkpoints – ‘catastrophe’ To finish: must get past all red checkpoints. Exponen?ally unlikely.
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E+S ES
L
Reac?on coordinate (a measure of reac?on progress)
Error correc'ng kine'c limit
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Reac'on coordinate & progress
L
%me
catastrophe Exponen?ally unlikely ->
Start Finish
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Catastrophes at checkpoints
e
∆d
ERi EWi
f d f
E+S ES
pR
- forw. =
f d + f > pW
- forw. =
f e∆d + f
Forward
η ⇠ pW
forw.
pR
forw.
!n ! e−n∆ when f ⌧ d, e∆d when f ⌧ d, e∆d
Need
Low error rate but very slow comple%on rate
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Catastrophes and rescues
L
%me
catastrophe Exponen?ally unlikely ->
Start Finish
rescue
E+S ES
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t L
unbounded bounded
fcat > fres fcat < fres
Catastrophes and rescues
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Reac'on coordinate & progress
η ∼ e−n∆, T ∼ Λn η ∼ e−n∆0, T ∼ nκ
η ∼ e−∆, T ∼ nγ
Lowest error, Highest ?me Low error, Low ?me High error, Low ?me
- pres. ⌧ pR
- cat. < pW
cat.
pR
- cat. < pres. < pW
cat.
pR
- cat. < pW
- cat. < pres.
R W Error rate Dissipa?on/Time
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Energy vs Error Rate tradeoff
E+S ES
D T D T # of fu?le cycles ~ # of ATP molecules used ~ dissipa?on D T
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Dynamic instability of microtubules
Tim Mitchison (HMS)
t L
unbounded bounded
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Non-equilibrium growth of microtubules
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Microtubule growth regimes
t L
unbounded bounded
fcat > fres fcat < fres
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Microtubule growth as a search strategy
chromosome microtubule Wrong direc3on Right direc3on Signaling molecule (lowers cat.) Proposal: Set <catastrophe rate> ~ <rescue rate> Bounded-unbounded transi?on point Bounded
- cat. > rescue
Unbounded
- cat. < rescue
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Kirschner/Gerhart Foraging ants Microtubules When should you return home and try again?
Microtubules and foraging
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Algorithms that get stuck
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Restarts cut tail of first passage ?me distribu?on Start Finish Stuck Stuck Stuck
Algorithms that get stuck
Example: Simulated annealing
- n a glassy landscape
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Restarts cut tail of first passage ?me distribu?on Start Right Finish Wrong finish Wrong finish Wrong finish
Duality: Errors vs (first passage) Time
Example: Simulated annealing
- n a glassy landscape
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Summary
Proofreading = Dynamic instability in chemical space (catastrophes and rescues) Happy medium in error-energy tradeoff Happy medium in returns home (search, stochas?c algorithms..)