9/23/2015 kitpc - membrane biophysics 1 modeling of actomyosin driven cell oscillations xiaoqiang...
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Modeling of Actomyosin Driven Modeling of Actomyosin Driven Cell OscillationsCell Oscillations
Xiaoqiang WangXiaoqiang Wang
Florida State Univ.Florida State Univ.
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OutlineOutline
BackgroundBackground
Facts determine vesicle shapeFacts determine vesicle shape
A mechanism for the oscillationA mechanism for the oscillation
Mathematical model and Phase field Mathematical model and Phase field formulationsformulations
Numerical experimentNumerical experiment
Future work and conclusionFuture work and conclusion
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MembranesMembranes
Cellular membranes are composed mostly of Cellular membranes are composed mostly of lipids.lipids.
Lipid has one polar (hydrophilic) head and one or Lipid has one polar (hydrophilic) head and one or more hydrophobic tails.more hydrophobic tails.
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Vesicle membranesVesicle membranes
Lipids form a bilayer structure which is a basic building Lipids form a bilayer structure which is a basic building block for all bio-membranes.block for all bio-membranes.Membranes are fluid-like: lipids have rapid lateral Membranes are fluid-like: lipids have rapid lateral movement and slowly flip-flop movement.movement and slowly flip-flop movement.
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Cell OscillationCell Oscillation
These cell oscillations are driven by actin and myosin dynamics.
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Fragments of L929 fibroblastsFragments of L929 fibroblasts
L929 fibroblasts
Centrifugation after microfilaments and
microtubules depolymerization
Cytoplast
Fragments
Nucleus [E. Paluch, M. Piel, J. Prost, M. Bornens, C. Sykes, Biophys. J., 89:724-733]
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Facts determine vesicle shapeFacts determine vesicle shape
Elastic bending energy, measured by the Elastic bending energy, measured by the bending curvatures of the surface.bending curvatures of the surface.
Osmotic pressureOsmotic pressure
Surface tensionSurface tension
Two components Two components
Line tension energy Line tension energy
Dynamics inside the cell membrane (actin Dynamics inside the cell membrane (actin filaments, microtubules).filaments, microtubules).
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Elastic bending energyElastic bending energy
The Elastic Bending The Elastic Bending Energy is determined Energy is determined by the surface by the surface curvatures:curvatures:
Lipid bilayer builds
k: bending rigidity
CC00: spontaneous curvature: spontaneous curvature
Helfrich W., Z. Naturforsch, 1973Helfrich W., Z. Naturforsch, 1973
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Osmotic Potential EnergyOsmotic Potential Energy
Osmotic pressure depends on the salty density Osmotic pressure depends on the salty density difference between inside and outside of the difference between inside and outside of the membrane. The osmotic pressure is proportional membrane. The osmotic pressure is proportional to density difference between inside and to density difference between inside and outside. outside.
In the case with zero outside density, the In the case with zero outside density, the osmotic pressure inverse proportional to the osmotic pressure inverse proportional to the inside volume i.e. inside volume i.e.
We formulate the osmotic potential energy byWe formulate the osmotic potential energy by
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Surface Tension and Line TensionSurface Tension and Line Tension
The oscillating cell membranes can be The oscillating cell membranes can be divided into two components, with different divided into two components, with different actin and myosin concentrations.actin and myosin concentrations.
Besides the Elastic Bending Energy, Besides the Elastic Bending Energy, different components has different surface different components has different surface tension, which can be formulated bytension, which can be formulated by
Line tension energy involves between the Line tension energy involves between the two components. It can be formulated by two components. It can be formulated by
or or
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Dynamic characterization of actin Dynamic characterization of actin during the oscillationduring the oscillation
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Dynamic characterization of myosin Dynamic characterization of myosin II during the oscillationII during the oscillation
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Cell cortex: Stress due to myosin Cell cortex: Stress due to myosin motorsmotors
-helical coiled coil
heavy chain motor domains
Myosin II
light chains
actin
actin
myosin
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A mechanism for the oscillationA mechanism for the oscillation
[E. Paluch, M. Piel, J. Prost, M. Bornens, C. Sykes, Biophys. J., 89:724-733]
Actin
Myosin
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Total EnergyTotal Energy
All together with the elastic bending energy, surface All together with the elastic bending energy, surface tension, line tension and osmotic potential of the lipid tension, line tension and osmotic potential of the lipid membrane, we have the total energymembrane, we have the total energy
where where ii are the surface tension coefficients, are the surface tension coefficients, ii are the are the bending rigidities, Cbending rigidities, Cii are the spontaneous curvatures. are the spontaneous curvatures.
The surface tension and spontaneous curvatures are The surface tension and spontaneous curvatures are depending on the density of myosin II of each depending on the density of myosin II of each component, and we setcomponent, and we set
where ywhere yii are the density values of myosin. are the density values of myosin.
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Lipids TransferLipids Transfer
The total number of the lipid molecules is fixed and the lipids may The total number of the lipid molecules is fixed and the lipids may move from one component to the other. move from one component to the other. Suppose the nature area of two components are ASuppose the nature area of two components are A11(t) and A(t) and A22(t), (t), written by the penalty formulation to the total energy:written by the penalty formulation to the total energy:
The interior surface tension / pressure is proportional to the The interior surface tension / pressure is proportional to the Lagrange terms, i.e. Lagrange terms, i.e.
The lipid moving rate from one component to the other is assumed The lipid moving rate from one component to the other is assumed to be proportional to the pressure difference, i.e.to be proportional to the pressure difference, i.e.
And we haveAnd we have
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Polymerization and Diffusion of Polymerization and Diffusion of ActinActin
The concentrations of actin and myosin II are different on different The concentrations of actin and myosin II are different on different membrane components. membrane components. Actin polymerization occurs at the surface ends whereas Actin polymerization occurs at the surface ends whereas depolymerization occurs at the pointed ends. The growth velocity of depolymerization occurs at the pointed ends. The growth velocity of the actin gel:the actin gel:
where and are the rate constants at two ends, is the where and are the rate constants at two ends, is the concentration of G-actin available for polymerization.concentration of G-actin available for polymerization.And we have the mass conservation:And we have the mass conservation:
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Diffusion of Myosin IIDiffusion of Myosin II
Myosin II is combined with actin, it disassembles Myosin II is combined with actin, it disassembles to the solvent as the depolymerization of actin to the solvent as the depolymerization of actin filaments. On the other hand, it attaches to the filaments. On the other hand, it attaches to the filaments at any position. filaments at any position.
where is the attaching rate of myosin and is where is the attaching rate of myosin and is the depolymerization of actin filaments, , the depolymerization of actin filaments, , and are the concentration of myosin II in and are the concentration of myosin II in solvent, component 1 and component 2. solvent, component 1 and component 2. Also the mass conservation:Also the mass conservation:
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Membrane Phase Field FunctionMembrane Phase Field Function
Introduce a phase function , defined Introduce a phase function , defined on a computational domain , to label the on a computational domain , to label the inside/outside of the vesicleinside/outside of the vesicle
Membrane : the level set Membrane : the level set
=0>0
<0
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Membrane Phase Field FunctionMembrane Phase Field Function
Ideal phase field functionIdeal phase field functiond: distance functiond: distance function
+1+1 inside, inside, -1-1 outside outside
Sharp interface as Sharp interface as !! 0 0
-1
1
=1
=-1
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Elastic Bending Energy in Phase Elastic Bending Energy in Phase Field ModelField Model
TakingTaking
On the other hand, On the other hand,
minimizing => minimizing =>
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Component Phase Field FunctionComponent Phase Field Function
phase field functionphase field functiond: distance functiond: distance function
+1+1 one component, one component, -1-1 another another
bending rigidity bending rigidity is a function of is a function of
=1
=-1
1
-1
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Phase Field FormulationsPhase Field Formulations
Surface tension:Surface tension:
where where Elastic bending energy:Elastic bending energy:
wherewhereLine tension energy with Line tension energy with
Osmotic potential energy with Osmotic potential energy with
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Phase Field FormulationsPhase Field Formulations
Perpendicular of Perpendicular of and and ? ? ::
Tanh profile preserving of Tanh profile preserving of ::
Total energy:Total energy:
System with gradient flow:System with gradient flow:
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Numerical SchemesNumerical Schemes
Axis-symmetric or truly 3D configurations.Axis-symmetric or truly 3D configurations.Spatial discretization:Spatial discretization:
Finite Difference and Fourier Spectral.Finite Difference and Fourier Spectral.Time discretization:Time discretization:Explicit Forward Euler / Implicit SchemesExplicit Forward Euler / Implicit SchemesTime step size is adjusted to ensure the gradient flow Time step size is adjusted to ensure the gradient flow part: part:
Update area AUpdate area A11(t), A(t), A22(t), actin concentrations m(t), actin concentrations m00, h, h11, h, h22, , myosin II concentrations ymyosin II concentrations y00, y, y11, y, y22 every time step after every time step after the gradient flow of the gradient flow of and and . .
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Numerical ResultsNumerical Results
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Numerical ResultsNumerical Results
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Future workFuture work
Simulations of the breakage
More numerical simulations for examining the effect of osmotic pressure, spontaneous curvature, line tension, etc.
Coupling with fluid
Reaction diffusion of actin and myosin
Improvement of our phase field formulations
More rigorous theoretical analysis of our models
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SummarySummary
We proposed a model together with the phase We proposed a model together with the phase field simulation to explain the oscillation of cell field simulation to explain the oscillation of cell membrane.membrane.Some preliminary analysis and numerical Some preliminary analysis and numerical simulations have been carried out and simulations have been carried out and compared with experiment findings.compared with experiment findings.The simulation results illustrate how the cell membrane interact with the interior actin dynamics, the competition of the surface tension, bending stiffness and the interfacial line tension. More studies are underway… More studies are underway…