reciprocity between robustness and plasticity as a universal quantitative law in biology - tetsuhiro...
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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
Robustness ßà Plasticity (Constancy)
Compatible at various levels
(Changeability)
Conflicting?
Robustness ßà Plasticity (Constancy)
Compatible at various levels
(Changeability)
Conflicting?
Is there some quantitative relations ? YES !!
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
Tradeoff Robustness --- Plasticity
Reciprocity Robustness --- Plasticity
Robustness --- Plasticity
Robustness --- Plasticity
Circadian clock (Temporal pattern formation)
TSH, Kaneko, PNAS (2012) TSH, Kaneko, FEBS Lett. (2014) TSH, Kaneko, Phys Rev Lett (2015)
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)
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
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)
Entrainment by temperature cycles
(Yoshida et al,. PNAS 2009) (Liu et al., Science 1998)
Cyanobacteria (in vitro Kai-clock)
Mold (Neurospora crassa)
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
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 ?
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
In vitro circadian clock
(Ilustrated by David Goodsell)
KaiA KaiC
KaiB (Nakajima et al., Science, 2005)
Mixing in a test tube
Cyanobacteria
KaiC phosphorylation cycle KaiA KaiB
Phosphorylation
Dephosphorylation
KaiC: Autokinase and autophosphatase KaiA: Facilitator for KaiC’s kinase activity KaiB: Inhibitor of KaiA
KaiC
KaiC Phosphorylation Model
Adapted from (van Zon, Lubensky, ten Wolde., PNAS 2007)
KaiC Phosphorylation Model
Competition of enzyme
Temperature compensation below Tc i×[Ci ]∑ / 6×[C]T
0
1
0 150Time (h)
0
1
Ratio
0
1
KaiC phosphorylationFree KaiA / Total KaiA
β = 1.0(High)
1.5
2.0(Low)
Decrease in amplitude below TC
0
1
0 150Time (h)
0
1
Ratio
0
1
KaiC phosphorylationFree KaiA / Total KaiA
β = 1.0(High)
1.5
2.0(Low)
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)
C0C4
C1C5
C2C6
C3
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
Speed of rate-limit reactions is compensated
For sufficient small [A]T
Afree Atotal
1+CmKm
AtotalKmCm
∝exp(βEp )
kpAfree ⇠ exp(��Ep) exp(�Ep)
Free enzyme as “Buffer Molecule”
⌃
˜C / exp(��(Ep � Edp))
Ci ⇠ kdp⌃ ˜C / exp(��Ep)
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)
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 ?
Entrainment
Temperature-compensated clock can be entrained by temperature cycles
More temperature-compensated clock shows faster entrainment
0.0
0.05
Entrainability
-0.10.0
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
Indicator of plasticity of phase
Changes in activation energy of dephosphorylation Amplitude of PRC à Δφ
Ep = 1.00 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
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
-0.030.0
0.18
-0.10.0
0.6
0.0 1.0EdpΔ
T / T Δ
φ
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.)
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?
Reciprocity between robustness and plasticity in TTO
Gene
mRNA (M)φ
Nucleus
φ
Proteinprecursor (R)
Protein (Q)Nucreic
protein (P)
k
a s
c b
duv
Kurosawa, Iwasa, JTB (2005)
-0.030.0
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
Reciprocity is independent of nonlinearity of model
van der Pol equation dx
dt
= e
��Eiy
dy
dt
= ✏(e��Ea � e
��Eix
2)y � e
��Eibx
-0.0030.0
0.018
-0.10.0
0.7
ΔT
/ T Δφ
ε = 0.1
0.0 1.0Ei
-0.030.0
0.24
-0.2
0.0
0.7
0.0 1.0Ei
ΔT
/ T Δφ
ε = 2.0Weak nonlinearity Strong nonlinearity
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”
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)
Stuart-Landau equation dR
dt= R�R3 (Amplitude)
(Angle)
Environment
Velocity
Period
Amplitudef1f2
àIncluding feed-forward adaptation
dR(�)
dt= f1(�)R�R3
d⇥(�)
dt= f1(�)! + f2(�)R
2
d⇥
dt= ! +R2
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
� lnT (�) = �� ln f1(�)�� ln f2(�)
� ln f1(�) = �� ln f2(�) , the period is robust When
Plasticity of phase �(R,⇥,�) = ⇥+ f2(�)
⇢lnR� 1
2ln 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
2ln f1(� +��)� 1
2ln f1(�)
�
a� lnT +�� = c
(a, c is constant independent of f1(β))
Reciprocity
Reciprocity relationship
Reciprocity is achieved by adaptation on the limit-cycle via a buffer molecule
Orbit before ennvironmental changeOrbit compensated perfectlyOrbit compensated partially
Concentration of buffer molecule, x
Conc
entra
tion
of o
ther
mole
cules
Δx Δx* - Δx
�T/T / �x
⇤ ��x
�� / �x
a�T
T+ b�� = c
TSH, Kaneko, PRL (2015)
(a, b, c = const.)
Robust cellular polarity and chemo- and thermotaxis
(Spatial pattern formation)
Unpublished
Cellular differentiation Single cell level plasticity and multi cell level robustness
Unpublished
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) … ?
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