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Correlated lateral phase separations in stacks of lipid membranes
- 1. Correlated lateral phase separations in stacks of lipid menbranes Takuma Hoshino M26 (積層脂質膜における相分離の連動性)
- 2. Biomembranes and stacked structures ๏ Biomembrane: Lipid bilayer + Proteins ๏ Stacked structure: energy synthesis, material transport Intercellular lipidsCorneocytes Mitochondria Stratum corneum of human skin Saturated lipid + Cholesterol Unsaturated lipid
- 3. Domain formation in multi-component membranes ๏ Unsaturated lipid + Saturated lipid + Cholesterol ๏ Liquid-disordered (Ld) phase + Liquid-ordered (Lo) phase One-phase Cholesterol Ld Lo Ld-phase: Unsaturated lipids Lo-phase: Saturated lipids + Cholesterols
- 4. Domain formation in stacked membranes ๏ Multi-component stacked membranes ๏ Vertical correlation through the inter-layer interaction L. Tayebi et al., Nat. Mater. 11, 1074 (2012) Substrate z Top view Domain connectivity in stacked membranes Acceleration of domain growth as compared to uni-lamellar case
- 5. Stacked 2-dimensional Ising model ๏ Binary membranes: Unsaturated + Saturated ๏ Hamiltonian Intra-membrane Inter-membrane
- 6. Monte Carlo simulations ๏ Lattice size: L×L×Lz ๏ Parameters: T/J, λ=J’/J ๏ Kawasaki exchange dynamics ๏ Lateral diffusion : =30:70: =50:50 High temperature → ,
- 7. ๏ Statics: phase diagram, domain connectivity ๏ Dynamics: domain growth exponent Aims (J’)*Connected Disconnected Statics Dynamics
- 8. ๏ Specific heat ๏ Finite-size scaling analysis → Tc (λ) Statics: specific heat Determining Tc as a function of λ Lz= 8 λ=0.1 Specificheat Tc (λ = 0.1)
- 9. ๏ Strong correlation in z-direction due to the cooperative behavior of domains ๏ Critical temperature as a function of λ Statics: phase diagram L=48λ=0.1 Specificheat Crossover from 2D to 3D 2D 3D
- 10. ๏ Internal energy and entropy ๏ Critical inter-layer interaction ๏ Statics: connectivity of domains ๏ Connectivity L=16 Lz=8 Domain connectivity in stacked membranes λ* Connectivity Disconnected Connected In thermodynamic limit, domains are always connected for J’ > 0
- 11. Summary of statics ๏ Critical temperature Tc as a function of λ ๏ Connected-disconnected transition
- 12. Dynamics: scaling law Scaling hypothesis ๏ Average domain size: ๏ Total domain area is conserved ๏ Total interface length: Binder and Stauffer theory ๏ Diffusion of domains ๏ Growth exponent ๏ Binder and Stauffer, Phys. Rev. Lett. 33, 1006 (1974) : Space dimension
- 13. ๏ Time evolution of domains ๏ Constant quench ratio (T=0.6Tc) ๏ L=256 Lz=8 Dynamics: domain growth (T=0.6Tc) MCS Crossover from 2D to 3D when the quench temperature is fixed Interfacelength Growthexponent
- 14. Growth exponent becomes larger with increasing λ Power-law↑ Growthexponent L=256 Lz=8 Interfacelength ๏ Time evolution of domains ๏ Constant temperature (T/J=2.0) Dynamics: domain growth (T/J=2.0) Acceleration of domain growth as compared to uni-lamellar case
- 15. Summary of dynamics ๏ Time evolution of domains ๏ Growth exponent L=256 Lz=8 Constant quench ratio (T=0.6Tc) Constant temperature (T/J=2.0) Growthexponent Power-law↑Growthexponent
- 16. Summary and future plan Summary ๏ Correlated lateral phase separation in stacked membranes ๏ Stacked 2d Ising model & MC simulations ๏ Phase diagram & Domain connectivity ๏ Domain growth-law Future work ๏ Stacked 2d Ising model with membrane fluctuations T. Hoshino, S. Komura, and D. Andelman, J. Chem. Phys. 143, 243124 (2015)
Editor's Notes
- 1m__1m Biological membranes play an important roll in living organisms. Biological membranes form bilayer structures, where the head groups in both monolayers face with a water-rich environment. Biomembranes mainly composed of phospholipids such as unsaturated lipid and saturated lipid but contain also other molecules such as cholesterol, glyco-sugar and protein. In living organisms, these membranes can form not only a plasma membrane but also multi-lamellar stacks known as lamellar bodies. Examples of these stacked structures are mitochondria and stratum corneum of human skin. It is thought that a stacked structure is related to energy synthesis and material transport. In order to mimic the role of plasma membranes
- 1m00s__2m In order to mimic the role of plasma membranes, many studies have been performed on ternary mixtures consisting of unsaturated lipid such as phosphatidylcholine, saturated lipid such as sphingomyelin and cholesterol. By decreasing temperature, these ternary mixtures undergo a lateral phase separation, where a liquid-disordered phase coexists with a liquid-ordered one. It is thought that these domains are relevant in a variety of cell-surface signaling in biological membranes. Although many researches have been performed on artificial membranes mimicking the role of plasma membranes, there are only a few studies on artificial stacked membranes.
- 1m__3m In 2012, Tayebi reported a experimental study on a stack of multicomponent lipid bilayers. It was observed that phase-separated domains are vertically connected across several hundred bilayers and form the columnar structure due to the strong interaction between bilayers. This columnar structure is shown in this picture taken by confocal microscopy. They found that this vertical correlation through the inter-layer interaction gives rise to domain connectivity in stacked membranes and acceleration of domain growth as compared to a uni-lamellar vesicle. Motivated by this study, we propose a theoretical model on stacked membranes.
- 1m00s__4m We consider binary membranes composed of unsaturated lipid and saturated lipid. The part mainly composed of unsaturated lipid and saturated lipid are evaluated as -1 and +1, respectively. The Hamiltonian of this system can be written as this equation. Here nearest neighbor spins within the layer interact with intra-layer interaction J and nearest neighbor spins between adjacent layers interact with inter-layer interaction J’. We perform Monte Carlo simulations by using this Hamiltonian.
- 1m__5m In our simulation, we define the lattice size as L times L times Lz and parameters are T divided by J and inter-layer coupling strength lambda. Here Kawasaki exchange dynamics is applied for our system. This dynamics allows nearest neighbor spins to exchange only within the layers. This exchange is corresponding to the lateral diffusion of lipid molecules. Examples of our simulation are shown in this conditions. In 50:50 compositions, the phase-separated domains are gradually connected along z-direction. This behavior can be seen in 30:70 compositions as well. In 30:70 compositions, domains form the columnar structure as shown in the experiment.
- 30s__5m30s From these simulations, we set two aims . First aim is to understand the statics, especially we would like to obtain the phase diagram by determining critical temperature and understand the domain connectivity. Second aim is the dynamics. In particular, we would like to obtain the domain growth exponent as time evolves.
- 40s__6m10s In order to determine the critical temperature and obtain the corresponding phase diagram, we compute the specific heat per lattice site. In figure (a), we plot specific heat c as a function of T for each lateral size L with constant Lz. For each size, we associate the peak position of the specific heat. Finite-size scaling analysis is then performed in order to determine the critical temperature as shown in figure (b). This procedure is performed for different inter-layer coupling.
- 1m20s__7m30 Next, we calculate c for each Lz with constant L. Even if Lz is increased, the peak position of specific heat is almost independent of Lz at least for Lz is larger than 8. This is because a strong correlation in z-direction due to the cooperative behavior of domains. By determining critical temperature, we obtain the corresponding phase diagram as shown in right figure. The system is in a one-phase state above the solid line. The system is in a phase-separated state below the solid line. We find that the value of critical temperature interpolates from 2D Ising value 2.26 to 3D value 4.5. It is the cross-over from 2D to 3D as inter-layer interaction increases.
- 1m30s__9m00s Next, we define the connectivity of domains delta^2. When the domains are connected along the z-direction, the value of delta^2 approaches to 1. Otherwise this value is very small when the phase-separated domains are uncorrelated. We plot this quantity delta^2 as a function of temperature for each lambda. We find that domains are vertically connected when the inter-layer coupling lambda is larger than 0.01. We consider this lambda* is the boundary between connected and disconnected domains. In order to discuss this transition, we compare internal energy and entropy. As a result, we observe that critical inter-layer interaction (J’)* can be written as this form. In small size such as L=16, (J’)* can be finite because of finite-size effect. If lateral size L is large enough, (J’)* is equal to 0. This means that in thermodynamic limit, domains are always connected for any finite inter-layer interaction. And this theoretical result explains the domain connectivity in stacked membranes.
- 30s__9m30s This is the summary of statics. We determine Tc as a function of lambda. And we discuss the connected-disconnected transition. This is final phase diagram. The system is in 1-phase and disconnected phase state above the solid line. Below the line, the system is in 2-phase and connected phase state except lambda=0. Only lambda=0, the system is in 2-phase but disconnected phase state.
- 1m__10m30s Now we discuss the dynamics. Under the assumption that scaling laws can be applied, the average domain size R is proportional to t^alpha. When the total domain area A is conserved, R is inversely proportional to the total interface length l(t). Hence, l(t) is proportional to t^-alpha. Since our Kawasaki exchange dynamics corresponds to the lateral diffusion of lipid molecules, Binder and Stauffer theory can be applied. In this theory, domains are composed of particles. Within a diffusion of domains, a growth exponent alpha can be written as 1 over d+3. Here d is space dimension. Hence, alpha is equal to 1/5 for 2D, 1/6 for 3D. In order to observe the cross-over from 2d to 3d,,,
- 1m__11m30s ,,,we keep a quench ratio T=0.6Tc. In left figure, we plot the total interface length l(t) as a function of time measured in Monte Carlo steps. Growth exponent alpha is extracted from late stage kinetics. In right figure, we plot the extracted growth exponent as a function of lambda. We find that the value of alpha is equal to 0.24 for lambda=0, 0.14 except lambda=0. As a result, we find cross-over from 2D to 3D when the quench temperature is fixed. Although this behavior is theoretically correct, the final temperature should be fixed in a experiment.
- 1m__12m30s In order to mimic the experimental condition, we keep the final temperature T/J=2.0, which means that the quench depth can be varied as inter-layer coupling is increased. In this case, we can not observe cross-over from 2D to 3D. However, the extracted growth exponent becomes larger with increasing lambda. This is because the quench depth becomes larger as lambda is increased. This result explains the acceleration of domain growth as compared to uni-lamellar case.
- 30s__13m00s This is summary of dynamics. We study time evolution of domains under scaling hypothesis. First we find cross-over from 2D to 3D as shown in left figure. Then we find that growth exponent becomes larger with increasing lambda as shown in right figure.
- 1m__14m00s This is summary. We study correlated lateral phase separation in stacked membranes. We propose a theoretical model and perform the MC simulations. In statics, we obtain the phase diagram and discuss domain connectivity. In dynamics, we calculate growth exponent and find exponent alpha becomes larger with increasing lambda. Now we are planning to expand this stacked 2d Ising model. Previously, the inter-layer interaction is constant because the membranes are flat. Now we consider this interaction can varied with membrane fluctuations.