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Reciprocity between robustness and plasticity as a universal quantitative law in biology - Tetsuhiro S. Hatakeyama

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Reciprocity between robustness and plasticity as a universal quantitative law in biology - Tetsuhiro S. Hatakeyama

  1. 1. Reciprocity between robustness and plasticity as a universal quantitative law in biology Tetsuhiro S. Hatakeyama The University of Tokyo Quantitative laws II @ Como 13. June. 16
  2. 2. Robustness ßà Plasticity (Constancy) Compatible at various levels (Changeability) Conflicting?
  3. 3. Robustness ßà Plasticity (Constancy) Compatible at various levels (Changeability) Conflicting? Is there some quantitative relations ? YES !!
  4. 4. Robustness ßà Plasticity •  Circadian clock –  Temporal pattern formation •  Robust cellular polarity and chemo- and thermotaxis –  Spatial pattern formation •  Cellular differentiation –  Single cell level plasticity and multi cell level robustness There is a reciprocity relationship
  5. 5. Tradeoff Robustness --- Plasticity Reciprocity Robustness --- Plasticity Robustness --- Plasticity Robustness --- Plasticity
  6. 6. Circadian clock (Temporal pattern formation) TSH, Kaneko, PNAS (2012) TSH, Kaneko, FEBS Lett. (2014) TSH, Kaneko, Phys Rev Lett (2015)
  7. 7. Criteria of the circadian rhythm 1.  The rhythm persists in constant condition with a period of 24 hours 2.  The rhythm exhibit temperature and nutrient compensation of period 3.  The rhythm can be entrained by external conditions (light/dark, temperature cycles)
  8. 8. Belousov- Zhabotinsky reaction (Dutt and Muller. J.Phys.Chem. 1993, Nakajima et al,. Science 2005) 25℃ 35℃ 0.3 minutes 0.15 minutes 50% In vitro cyanobacterial circadian clock 22 hours 21 hours 95% Temperature compensation
  9. 9. Criteria of the circadian rhythm 1.  The rhythm persists in constant condition with a period of 24 hours 2.  The rhythm exhibit temperature and nutrient compensation of period à Robustness of the period 3.  The rhythm can be entrained by external conditions (light/dark, temperature cycles)
  10. 10. Entrainment by temperature cycles (Yoshida et al,. PNAS 2009) (Liu et al., Science 1998) Cyanobacteria (in vitro Kai-clock) Mold (Neurospora crassa)
  11. 11. Criteria of the circadian rhythm 1.  The rhythm persists in constant condition with a period of 24 hours 2.  The rhythm exhibit temperature and nutrient compensation of period à Robustness of the period 3.  The rhythm can be entrained by external conditions (light/dark, temperature cycles) à Plasticity of the phase
  12. 12. Criteria of the circadian rhythm 1.  The rhythm persists in constant condition with a period of 24 hours 2.  The rhythm exhibit temperature and nutrient compensation of period à Robustness of the period 3.  The rhythm can be entrained by external conditions (light/dark, temperature cycles) à Plasticity of the phase Is there some relations ?
  13. 13. Two mechanisms of circadian clocks Post-translational oscillator (PTO) – Oscillation is generated by protein-protein interactions (w/o transcription, translation) Transcription-translation-based oscillator (TTO) – Oscillation is generated by negative feedback loops by transcription and translation
  14. 14. In vitro circadian clock (Ilustrated by David Goodsell) KaiA KaiC KaiB (Nakajima et al., Science, 2005) Mixing in a test tube Cyanobacteria
  15. 15. KaiC phosphorylation cycle KaiA KaiB Phosphorylation Dephosphorylation KaiC: Autokinase and autophosphatase KaiA: Facilitator for KaiC s kinase activity KaiB: Inhibitor of KaiA KaiC
  16. 16. KaiC Phosphorylation Model Adapted from (van Zon, Lubensky, ten Wolde., PNAS 2007)
  17. 17. KaiC Phosphorylation Model Competition of enzyme
  18. 18. Temperature compensation below Tc i×[Ci ]∑ / 6×[C]T 0 1 0 150Time (h) 0 1 Ratio 0 1 KaiC phosphorylation Free KaiA / Total KaiA β = 1.0 (High) 1.5 2.0 (Low)
  19. 19. Decrease in amplitude below TC 0 1 0 150Time (h) 0 1 Ratio 0 1 KaiC phosphorylation Free KaiA / Total KaiA β = 1.0 (High) 1.5 2.0 (Low)
  20. 20. Accumulation of some forms of KaiC β = 1.0 (High) 1.5 2.0 (Low) 0 1 0 1 Ratio 0 1 0 150Time (h) C0 C4 C1 C5 C2 C6 C3
  21. 21. Intuitive explanation of temperature compensation At the low temperature, amount of KaiC that go round circuit decreases à Competition for enzyme is weakened Free enzyme works as a buffer Afreekp
  22. 22. Speed of rate-limit reactions is compensated For sufficient small [A]T Afree  Atotal 1+ Cm Km  Atotal Km Cm ∝exp(βEp ) kpAfree ⇠ exp( Ep) exp( Ep) Free enzyme as Buffer Molecule ⌃ ˜C / exp( (Ep Edp)) Ci ⇠ kdp⌃ ˜C / exp( Ep)
  23. 23. Two conditions for temperature compensation •  Amount of the enzyme is sufficiently small •  Difference in temperature dependence between phosphorylation and dephosphorylation (Different activation energies)  àWhen phosphorylation is rate-limiting, temperature compensation is achieved TSH, Kaneko, PNAS (2012) TSH, Kaneko, FEBS Lett. (2014)
  24. 24. Criteria of the circadian rhythm 1.  The rhythm persists in constant condition with a period of 24 hours 2.  The rhythm exhibit temperature and nutrient compensation of period à Robustness of the period 3.  The rhythm can be entrained by external conditions (light/dark, temperature cycles) à Plasticity of the phase Is there some relations ?
  25. 25. Entrainment Temperature-compensated clock can be entrained by temperature cycles
  26. 26. More temperature-compensated clock shows faster entrainment 0.0 0.05 Entrainability -0.1 0.0 0.6 ΔT/T 0.0 1.0Edp ΔT / T : (T(β2 ) - T(β1 )) / T(β1 ) Entrainability Entrainability --- Inverse of time for the perfect entrainment by external temperature cycles. Entrainability depends on the shape of external cycles à better indicator is needed
  27. 27. Indicator of plasticity of phase Changes in activation energy of dephosphorylation Amplitude of PRC à Δφ Ep = 1.0 0 π 2π 0 0.04π -0.14π Edp = 0.0 0.2 0.4 0.6 0.8 1.0 Δφ Phase response curve (PRC) against temperature pulse
  28. 28. More robust oscillation is more plastic !! Ep = 1.0 0 π 2π 0 0.04π -0.14π Edp = 0.0 0.2 0.4 0.6 0.8 1.0 -0.03 0.0 0.18 -0.1 0.0 0.6 0.0 1.0Edp ΔT/T Δφ ΔT / T : (T(β2 ) - T(β1 )) / T(β1 ) Δφ : Normalized amplitude of PRC Reciprocity between robustness and plasticity in PTO a T T + b = c (a, b, c = const.)
  29. 29. Two mechanisms of circadian clocks Post-translational oscillator (PTO) – Oscillation is generated by protein-protein interactions (w/o transcription, translation) Transcription-translation-based oscillator (TTO) – Oscillation is generated by negative feedback loops by transcription and translation Does reciprocity depend on mechanisms?
  30. 30. Reciprocity between robustness and plasticity in TTO Gene mRNA (M) φ Nucleus φ Protein precursor (R) Protein (Q) Nucreic protein (P) k a s c b du v Kurosawa, Iwasa, JTB (2005) -0.03 0.0 0.24 -0.2 0.0 0.7 0.0 1.0Ei ΔT/T Δφ ΔT / T : (T(β2 ) - T(β1 )) / T(β1 ) Δφ : Normalized amplitude of PRC 0 π 2π 0 0.1π -0.2π Ei = 0.0 0.2 0.4 0.6 0.8 1.0
  31. 31. Reciprocity is independent of nonlinearity of model van der Pol equation dx dt = e Ei y dy dt = ✏(e Ea e Ei x2 )y e Ei bx -0.003 0.0 0.018 -0.1 0.0 0.7 ΔT/T Δφ ε = 0.1 0.0 1.0Ei -0.03 0.0 0.24 -0.2 0.0 0.7 0.0 1.0Ei ΔT/T Δφ ε = 2.0 Weak nonlinearity Strong nonlinearity
  32. 32. General mechanism of robustness of the period Environment Period Buffer Molecules ( Amplitude) Rate-limit reaction ( Angular velocity) Input x Output y Robustness of the period is considered as adaptation on the limit-cycle
  33. 33. Intuitive explanation of reciprocity relationship Velocity is altered Amplitude is also altered àPhase is altered by amplitude Strong adaptation à Change in period ↓ Change in phase↑ Weak adaptation à  Change in period ↑ Change in phase ↓ Environment Period Buffer Molecules ( Amplitude) Rate-limit reaction ( Angular velocity)
  34. 34. Stuart-Landau equation dR dt = R R3 (Amplitude) (Angle) Environment Velocity Period Amplitude f1 f2 àIncluding feed-forward adaptation dR( ) dt = f1( )R R3 d⇥( ) dt = f1( )! + f2( )R2 d⇥ dt = ! + R2
  35. 35. Robustness of period Amplitude is altered by beta R⇤ ( ) = (f1( ))1/2 Angular velocity is also altered d⇥( ) dt = f1( )f2( ) Change in the period à T( ) = 2⇡(f1( )f2( )) 1 ln T( ) = ln f1( ) ln f2( ) ln f1( ) = ln f2( ) , the period is robustWhen
  36. 36. Plasticity of phase (R, ⇥, ) = ⇥ + f2( ) ⇢ ln R 1 2 ln f1( ) Transient change in β from β to β+Δβ (Amplitude is changed, but angle is not) Then, for any f1(β), reciprocity is achieved ( ) = f2( ) ln f1( )/2 ( + ) = ⇥( ) + f2( ) ⇢ 1 2 ln f1( + ) 1 2 ln f1( ) a ln T + = c (a, c is constant independent of f1(β)) Reciprocity
  37. 37. Reciprocity relationship Reciprocity is achieved by adaptation on the limit-cycle via a buffer molecule Orbit before ennvironmental change Orbit compensated perfectly Orbit compensated partially Concentration of buffer molecule, x Concentrationof othermolecules Δx Δx* - Δx T/T / x⇤ x / x a T T + b = c TSH, Kaneko, PRL (2015) (a, b, c = const.)
  38. 38. Robust cellular polarity and chemo- and thermotaxis (Spatial pattern formation)
  39. 39. Unpublished
  40. 40. Cellular differentiation Single cell level plasticity and multi cell level robustness
  41. 41. Unpublished
  42. 42. There is reciprocity •  Circadian clock –  Temporal pattern formation •  Robust cellular polarity and chemo- and thermotaxis –  Spatial pattern formation •  Cellular differentiation –  Single cell level plasticity and multi cell level robustness •  Evolution…? à Kaneko s talk (Next week) … ?
  43. 43. Take home message ○○ is robust ↓ There will be plastic conjugate properties ↓ Reciprocity will be held !! Everything needs to change, so everything can stay the same. ̶ Giuseppe Tomasi di Lampedusa, The Leopard

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