Wave Pinning, Actin Waves, and LPA
Transcript of Wave Pinning, Actin Waves, and LPA
![Page 1: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/1.jpg)
Wave Pinning, Actin Waves,
and LPA
MCB 2012William R. Holmes
Monday, 14 May, 12
![Page 2: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/2.jpg)
Intercellular Waves
• Dynamic Hem1 waves in neutrophils
Weiner et al., 2007, PLoS Biology
Monday, 14 May, 12
![Page 3: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/3.jpg)
Questions?
• How can such waves / pulses form?
• What molecular constituents play a role?
Monday, 14 May, 12
![Page 4: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/4.jpg)
FitzHugh Nagumo Idea
• Positive feedback induces a wave.
• Slower negative feedback yields a pulse.
Monday, 14 May, 12
![Page 5: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/5.jpg)
FitzHugh Nagumo
• Used to describe signal propagation in nerves.
• v = membrane voltage
• w = ion concentration
vt = v � v3 � w + I +Dv4v,
⌧wt = v � bw � a
Monday, 14 May, 12
![Page 6: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/6.jpg)
FitzHugh Nagumo
• Stable HSS leads to transient dynamics.
Stable HSS
v-nullcline
w-nullcline
Monday, 14 May, 12
![Page 7: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/7.jpg)
FitzHugh Nagumo
• Unstable HSS leads to a limit cycles and persistent dynamics
Unstable HSSw-nullcline
Monday, 14 May, 12
![Page 8: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/8.jpg)
FN Feature
• Stable HSS = transient excitable dynamics
• Unstable HSS = unstable persistent dynamics
Monday, 14 May, 12
![Page 9: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/9.jpg)
Relationship to cell dynamics
• Cells are not always all or nothing.
• Some cells exhibit dynamics even after a stimulus is removed.
• Some cells are excitable and persistent.
Monday, 14 May, 12
![Page 10: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/10.jpg)
FN Extionsion
• We will consider an augmentation of the standard FN framework.
Monday, 14 May, 12
![Page 11: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/11.jpg)
Hypothesis
• We still consider a basic ‘wave generator + refractory feedback’ model.
• Wave Generator = Actin Regulators
• Refractory feedback = Actin
Monday, 14 May, 12
![Page 12: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/12.jpg)
Hypothesis
• Polarity proteins such as GTPases or Phosphoinositides serve as a wave generator.
• Actin polymerization inactivates these proteins acting as a refractory feedback.
Monday, 14 May, 12
![Page 13: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/13.jpg)
Wave Generator
• Consider a wave pinning (WP) model indicative of GTPase function.
Monday, 14 May, 12
![Page 14: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/14.jpg)
Wave Generator : GTPases
• NPF = actin nucleating protein
• Exists in 2 forms.
• Only the active NPF nucleates F-Actin
Active NPFInactive NPF
Monday, 14 May, 12
![Page 15: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/15.jpg)
Model Features
• Primary features
• Slow Fast Diffusion
• Autocatalysis
• Conservation
Monday, 14 May, 12
![Page 16: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/16.jpg)
Wave Pinning: Equations
f(u, v) = v(k0 +�un
Kn + un)� �u
ut(x, t) = f(u, v) +Du4u
vt(x, t) = �f(u, v) +Dv4v
u
vDu ⌧ Dv
Monday, 14 May, 12
![Page 17: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/17.jpg)
LPA Reductionult(t) = f(ul, vg)
ugt (t) = f(ug, vg)
vgt (t) = �f(ug, vg)
ulug
vg
ug
Monday, 14 May, 12
![Page 18: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/18.jpg)
Conservation Reduction
• Assume the perturbation is highly localized.
Zu
g(t) + v
g(t) dx ⇡Z
u(x, t) + v(x, t) dx = C
• So vg(t) = T � ug(t)
Monday, 14 May, 12
![Page 19: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/19.jpg)
LPA System
ulug ug
ult(t) = f(ul, T � ug)
ugt (t) = f(ug, T � ug)
Monday, 14 May, 12
![Page 20: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/20.jpg)
Wave Pinning LPA
Local Branch
Well MixedBranch
Monday, 14 May, 12
![Page 21: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/21.jpg)
Wave Pinning: Stability
Stable HSS Stable HSS
Turing Unstable
Monday, 14 May, 12
![Page 22: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/22.jpg)
Wave Pinning: Stability
Stable HSS Stable HSS
Turing Unstable
Threshold
Monday, 14 May, 12
![Page 23: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/23.jpg)
Wave Pinning: Stability
Monday, 14 May, 12
![Page 24: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/24.jpg)
Wave Pinning
• + feedback yields a threshold response
• Conservation causes stalling.
• As the wave propagates, it depletes the inactive NPF
Monday, 14 May, 12
![Page 25: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/25.jpg)
Wave Pinning:Simulations
Wave Pinning+ Perturbation
Wave Pinning- Perturbation
TuringNoise
Active NPF
Monday, 14 May, 12
![Page 26: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/26.jpg)
Wave Pinning:Wave Generator
• We will assume this WP model acts as a wave generator.
Monday, 14 May, 12
![Page 27: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/27.jpg)
Refractory Feedback
• Polarity related proteins (GTPases) nucleate actin and initiate a wave.
• Growing actin ‘inactivates’ these proteins.
Monday, 14 May, 12
![Page 28: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/28.jpg)
Actin Wave Model
• Active NPF promotes F-Actin.
• Wave Generator
• F-Actin inactivates NPF
• Refractory feedback
NegativeFeedback
Monday, 14 May, 12
![Page 29: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/29.jpg)
Actin Wave ModelAt = f +DA4A,
It = �f +DI4I
f(A, I, F ) =
✓k0 +
�A3
A30 +A3
◆I � �
✓s1 + s2
F
F0 + F
◆A
Ft = ✏hh(A,F ) = knA� ksF
A
I
F
NPF Equations
F-Actin Equations
Monday, 14 May, 12
![Page 30: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/30.jpg)
Pulse Snapshot
• F-Actin wave trails NPF wave
Monday, 14 May, 12
![Page 31: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/31.jpg)
Spatio-TemporalBehaviour
• Kymograph = (x,t) plot
Pinned Wave Oscillating Wave
Reflecting Pulse
SinglePulse
PulseTrain Exotic
Monday, 14 May, 12
![Page 32: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/32.jpg)
Use LPA to map parameter space
Monday, 14 May, 12
![Page 33: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/33.jpg)
Actin Wave LP-System
Alt = f(Al, Ig, F l),
Agt = f(Ag, Ig, F g),
Igt = �f(Ag, Ig, F g),
F lt = ✏h(Al, F l),
F gt = ✏h(Ag, F g)
Monday, 14 May, 12
![Page 34: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/34.jpg)
Actin Wave LP-System
Alt = f(Al, Ig, F l),
Agt = f(Ag, Ig, F g),
Igt = �f(Ag, Ig, F g),
F lt = ✏h(Al, F l),
F gt = ✏h(Ag, F g)
Monday, 14 May, 12
![Page 35: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/35.jpg)
Actin Wave LPA
• Applying NPF conservation.
Alt = f(Al, C �Ag, F l),
Agt = f(Ag, C �Ag, F g),
F lt = ✏h(Al, F l),
F gt = ✏h(Ag, F g)
Monday, 14 May, 12
![Page 36: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/36.jpg)
Actin Wave LPA
H
H
H=Hopf Bifurcation • Branch Points are
retained from wave pinning.
• Hopf bifurcations are new and indicate oscillations.
Monday, 14 May, 12
![Page 37: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/37.jpg)
Actin Wave LPA
H
H
Oscillatory Instability
• Branch Points are retained from wave pinning.
• Hopf bifurcations are new and indicate oscillations.
Stable HSS
ExcitableOscillations
Monday, 14 May, 12
![Page 38: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/38.jpg)
Questions?
• How do the positive and negative feedback loops interact to initiate patterning?
• What role do they play in determining the resulting behaviour on a longer time scale?
Monday, 14 May, 12
![Page 39: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/39.jpg)
LPA + Simulation
• Curve = 2 parameter Hopf continuation.
• Points = values used for PDE simulations.
Monday, 14 May, 12
![Page 40: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/40.jpg)
LPA + Simulation
+ = Reflecting Waves
O = Reflecting Waves
* = Wave Trains
X = Single Wave
= Single Wave
Monday, 14 May, 12
![Page 41: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/41.jpg)
LPA + Simulation
• Inside the fish tail, patterning arises from instability.
• Outside, from excitability.
Monday, 14 May, 12
![Page 42: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/42.jpg)
Patterning Region
• Feedback is necessary for patterning, but to much suppresses it.
Monday, 14 May, 12
![Page 43: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/43.jpg)
Static to Dynamic Transition
• Increasing feedback yields a progression from static, to dynamic, to no patterning.
Monday, 14 May, 12
![Page 44: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/44.jpg)
Wave Trains
• Wave trains, indicative of target waves (in 2D), only occur inside the fish tail
(f)
Monday, 14 May, 12
![Page 45: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/45.jpg)
Conclusions
• GTPase like kinetics coupled with F-actin feedback is capable of producing a wealth of static and dynamic behaviours.
Monday, 14 May, 12
![Page 46: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/46.jpg)
Conclusions
Monday, 14 May, 12
![Page 47: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/47.jpg)
Conclusions
• The inclusion of a WP model (ie. conservation) as the ‘wave generator’ in the FitzHugh Nagumo framework yields substantially different behaviour.
Monday, 14 May, 12
![Page 48: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/48.jpg)
Conclusions
• The inclusion of a WP model (ie. conservation) as the ‘wave generator’ in the FitzHugh Nagumo framework yields substantially different behaviour.
• Static to dynamic transition.
Monday, 14 May, 12
![Page 49: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/49.jpg)
Conclusions
• The inclusion of a WP model (ie. conservation) as the ‘wave generator’ in the FitzHugh Nagumo framework yields substantially different behaviour.
• Static to dynamic transition.
• Reflecting waves vs wave trains.
Monday, 14 May, 12
![Page 50: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/50.jpg)
Conclusions
• The inclusion of a WP model (ie. conservation) as the ‘wave generator’ in the FitzHugh Nagumo framework yields substantially different behaviour.
• Static to dynamic transition.
• Reflecting waves vs wave trains.
• Persistent patterning in excitable regimes.
Monday, 14 May, 12
![Page 51: Wave Pinning, Actin Waves, and LPA](https://reader034.fdocuments.net/reader034/viewer/2022052613/628df3a1906a151e203675d7/html5/thumbnails/51.jpg)
Conclusions
• Increasing levels of feedback lead to a transition from static to dynamic behaviour and finally to the suppression of all patterning.
Monday, 14 May, 12