reciprocity between robustness and plasticity as a universal quantitative law in biology - tetsuhiro...

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