QUANTITATIVE LAWS June 13 -June 24

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. Robustness ßà Plasticity
(Constancy)
Compatible at various levels
(Changeability)
Conflicting?

3. Robustness ßà Plasticity
(Constancy)
Compatible at various levels
(Changeability)
Conflicting?
Is there some quantitative relations ?
YES !!

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. Tradeoff
Robustness --- Plasticity
Reciprocity
Robustness --- Plasticity
Robustness --- Plasticity
Robustness --- Plasticity

6. Circadian clock
(Temporal pattern formation)
TSH, Kaneko, PNAS (2012)
TSH, Kaneko, FEBS Lett. (2014)
TSH, Kaneko, Phys Rev Lett (2015)

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. 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. 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. Entrainment by
temperature cycles
(Yoshida et al,. PNAS 2009) (Liu et al., Science 1998)
Cyanobacteria
(in vitro Kai-clock)
Mold
(Neurospora crassa)

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. 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. 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. In vitro circadian clock
(Ilustrated by David Goodsell)
KaiA
KaiC
KaiB
(Nakajima et al., Science, 2005)
Mixing in
a test tube
Cyanobacteria

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. KaiC Phosphorylation Model
Adapted from (van Zon, Lubensky, ten Wolde., PNAS 2007)

17. KaiC Phosphorylation Model
Competition of enzyme

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. 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. 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. 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. 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. 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. 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. Entrainment
Temperature-compensated clock
can be entrained by temperature cycles

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Robust cellular polarity
and chemo- and thermotaxis
(Spatial pattern formation)

39. Unpublished

40. Cellular differentiation
Single cell level plasticity and
multi cell level robustness

41. Unpublished

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. 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