Phase Separation
Curvature Induced Phase Separation in the Golgi
Apparatus
Ross Magi
Department of Mathematics
University of Utah
May 17, 2013
Ross Magi
Curvature Induced Phase Separation in the Golgi Apparatus
Outline
of the mosaic
ways. Although
integral proteins (perhaps
nature of Model
this diversification
in particularinstances
a structure
attached to oli- theBiology
representedin Fig. 2, it is not
Phase Separation
Results is Future
Work
gosaccharides to form glycoproteins, matterof speculation,it is importantto indicatedwhetherit is the proteinor the
or interactingstronglywith specific lip- recognizethat the mosaic structureneed phospholipidthat providesthe matrixof
ids to form lipoproteins)alternatewith not be restrictedby the schematic rep- the mosaic. In other words, which comsections of phospholipidbilayer in the resentation in Fig. 2. Protein-protein ponent is the mortar,which the bricks?
cross section of the membrane(Fig. 2). interactionsthat are not explicitly con- This question must be answered when
The globularprotein molecules are pos- sidered in Fig. 2 may be important in the third dimensionof the mosaic structulated to be amphipathic(3, 4) as are determiningthe propertiesof the mem- ture is specified. Trhesetwo types of
the phospholipids. That is, they are brane. Such interactions may result mosaic structure may be expected to
structurallyasymmetric,with one highly either in the specific binding of a have very differentstructuraland funcpolar end and one nonpolar end. The peripheral protein to the exterior ex- tional properties, and the question is
highly polar region is one in which the posed surface of a particular integral therefore a critical one. It is our hyionic amino acid residues and any covalently bound saccharide residues are
clustered, and which is in contact with
the aqueous phase in the intact membrane;the nonpolarregion is devoid of
ionic and saccharideresidues, contains
many of the nonpolar residues, and is
embedded in the hydrophobic interior
of the membrane. The amphipathic
structure adopted by a particular integral protein (or lipoprotein)molecule,
and therefore the extent to which it is
embedded in the membrane,are under
thermodynamic control; that is, they
are determined by the amino acid sequence and covalent structure of the
protein, and by its interactionswith its
molecularenvironment,so that the free
energy of the system as a whole is at a
minimum.An integralprotein molecule
with the appropriatesize and structure,
or a suitable aggregateof integral proteins (below) may transversethe entire
membrane (3); that is, they have regions in contact with the aqueous sol- Fig. 3. The lipid-globularproteinmosaic model with a lipid matrix (the fluid mosaic
vent on both sides of the membrane. model); schematic three-dimensional and cross-sectional views. The solid bodies with
surfaces represent the globular integral proteins, which at long range are
J Singer
and Garth L Nicolson. The Fluid Mosaic
It is clear from these considerations Sstippled
randomly distributed in the plane of the membrane. At short range, some may form
that differentproteins, if they have the Model
as shown. of
In Cell
cross Membranes.
the Structure
Science,
section and in other
specific of
aggregates,
the legend of
details,New
Fig. 2 applies.
appropriate amino acid sequence to Series,
175(4023):720731, February 1972
Biology overview
Model Derivation
Numerical Results
18 FEBRUARY 1972
723
Ross Magi
Curvature Induced Phase Separation in the Golgi Apparatus
Phase Separation
Biology Model Results Future Work
The Golgi Apparatus
ORGANIZING POTENTIAL OF SPHINGOLIPIDS
17
Made of flattened membrane sacks
called cisternae
Important for modifying and
sorting proteins and lipids
Location of sphingolipid synthesis
Sphingolipids and glycerolipids
can phase separate
Phase separation may help with
protein sorting
FIG. 4. Model depicting how sphingolipid synthesis may contribute to the sorting power of the Golgi. As Golgi
cisternae mature (large gray arrow), sphingolipids are synthesized and gradually accumulate by being specifically
excluded from the tightly curved membrane in percolating COPI vesicles or tubular connections. By attracting endoplasmic reticulum (ER)-synthesized cholesterol, the flat sphingolipid-rich regions in the cisternal bilayer grow thicker
(see footnote 4). Due to their short transmembrane segments, Golgi enzymes and escaped ER-resident proteins are
excluded from the thick, sphingolipid/sterol-rich membrane regions where transport intermediates depart for the cell
surface. Instead, they will partition preferentially into earlier cisternae. Upon thickening of the bilayer, basolateral (or
lysosomal) membrane proteins adopt a conformation recognized by adaptor protein complexes (AP1/AP3; see footnote
30). AP-dependent removal of basolateral (lysosomal) material represents the terminal step in the cisternal maturation
process, leaving behind an apical membrane carrier whose content is determined by a “sorting by retention” principle
based on coaggregative properties of apical sphingolipids and proteins. Note that although the model implies a gradual
increase in membrane thickness across the Golgi stack, the shape of this thickness gradient is not known (see footnote
28). See text for further details. VTC, vesicular-tubular clusters; PGC, post-Golgi containers; PM, plasma membrane.
J.C.M. Holthuis, T. Pomorski, R.J. Raggers, H. Sprong,
and G van Meer. The organizing potential of
sphingolipids in intracellular membrane transport.
Physiological reviews, 81(4):16891723, 2001
transport (196, 310). As cargo moves through the stack, it
is modified by Golgi-associated processing enzymes.
These enzymes, which include numerous glycosidases
and glycosyltransferases, are generally not distributed
evenly between the cisternae, but often found in the order
in which they act on their substrates (see sect. VA).24 An
advantage of this compartmental organization is that
cargo can be exposed to an ordered array of processing
steps, allowing the cell to generate highly complex glycoproteins and glycosphingolipids. Positioned at the transside of the Golgi stack is the TGN. Here, processed cargo
is sorted, packaged into distinct vesicles (or larger me
brane carriers), and shipped to various destinatio
These post-Golgi destinations include the cell surfa
(either the apical or basolateral surface of epithe
cells), secretory storage granules, and the various co
partments of the endosomal/lysosomal system. The TG
not only serves as a major branching point in the sec
tory pathway but also forms the major site where
secretory and endocytic pathways become interconnect
This interconnection enables cells to balance membra
flow between the pathways and to maintain the prop
composition of their surfaces and intracellular organell
Goal: Build a model to explore the interplay between membrane shape
and lipid motion
Ross Magi
Curvature Induced Phase Separation in the Golgi Apparatus
24
It should be noted that this intra-Golgi separation is not precise
because the enzymes are generally spread over several cisternae, dis-
Phase Separation
Biology Model Results Future Work
The Model
Two-phase fluid model of a single Golgi cisterna
Two phases
Sphingolipids: volume fraction θs , velocity vs , chemical
potential µs , and more resistant to curvature
Glycerolipids: volume fraction θg , velocity vg , chemical
potential µg , and less resistant to curvature
Each phase behaves as a fluid and follows Navier-Stokes type
equations
Fluid motion is affected by membrane shape
Ross Magi
Curvature Induced Phase Separation in the Golgi Apparatus
Phase Separation
Biology Model Results Future Work
Model Equations
Constraint
θs + θg = 1
Continuity
∂θs
+ ∇ · (θs vs ) = 0
∂t
∂θg
+ ∇ · (θg vg ) = 0
∂t
Incompressibility
∇ · (θs vs + θg vg ) = 0
Force Balance
∇ · (θs σs ) − ξθs θg (vs − vg ) − θs ∇µs − θs ∇P = 0
∇ · (θg σg ) − ξθs θg (vg − vs ) − θg ∇µg − θg ∇P = 0
Ross Magi
Curvature Induced Phase Separation in the Golgi Apparatus
Phase Separation
Biology Model Results Future Work
Shape Equations
Let φ(x) define cisternal shape
Curvature, κ(x) =
dφ
dx
Constraints
Z 1
Z
sin(φ)dx =
1
0
cos(φ)dx = 0
0
Boundary Conditions
Φ(x)
φ(0) = φ(1) − 2π
φ0 (0) = φ0 (1)
Euler-Lagrange Equation
d
d
−
2g (θs , θg ) φ(x) −λ cos(φ(x))+µ sin(φ(x)) = 0
dx
dx
Ross Magi
Curvature Induced Phase Separation in the Golgi Apparatus
Phase Separation
Biology Model Results Future Work
Free Energy and Chemical Potential
Helmholz free energy from lattice model
F = U − TS
Low χsg
High χsg
Chemical Potentials
Free Energy Density
∂F
∂F
= µs ,
= µg
∂Ns
∂Ng
Interaction energy depends on curvature
ij = 0ij + cij κ2
Free Energy Density
0
0.5
1
θs
0
0.5
1
0
0.2
0.4
θs
0.6
0.8
f c (θs , θg ) = f (θs , θg ) + g (θs , θg )κ2
f (θs , θg ) = 0ss θs + 0gg θg + kb T χ0sg θs θg + kb T (θs ln θs + θg ln θg )
g (θs , θg ) = css θs + cgg θg + kb T χcsg θs θg
Ross Magi
Curvature Induced Phase Separation in the Golgi Apparatus
1
Phase Separation
Biology Model Results Future Work
Nondimensional 1D Equations
Continuity
∂θs
∂
+
(θs θg v ) = 0
∂t
∂x
Force Balance
∂
∂
∂
∂
ηθg
θs (θg v ) + θs
θg (θs v )
∂x
∂x
∂x
∂x
∂
−αθs θg v − βθs θg ((f c (θs ))0 ) = 0
∂x
Shape
d
−
dx
d
2g (θs , θg ) φ(x) − λ cos(φ(x)) + µ sin(φ(x)) = 0
dx
where θs + θg = 1 and v = vs − vg
Ross Magi
Curvature Induced Phase Separation in the Golgi Apparatus
Phase Separation
Biology Model Results Future Work
Numerical Implementation
Discretize system on staggered grid
θs , θg , and κ at cell centers
v and φ at cell edges
Numerical Algorithm:
1
2
Solve Euler-Lagrange equation for shape using Newton’s
method
Solve force balance equation for v using finite difference
method
Periodic boundary conditions
3
Step continuity equation in time using upwind method for
advection and Crank-Nicolson for diffusion
Periodic boundary conditions
Ross Magi
Curvature Induced Phase Separation in the Golgi Apparatus
Phase Separation
Biology Model Results Future Work
Phase Separation without Curvature
Snapshots in time with inital condition θs = 0.5(1 + 0.1 cos(4πx))
(a)
(b)
0.9
0.8
f (arbitrary units of energy)
0.7
0.6
θs
0.5
0.4
0.3
0.2
0.1
0
0.2
0.4
0.6
0.8
1
x
Ross Magi
0
0.2
0.4
θs
0.6
0.8
1
Curvature Induced Phase Separation in the Golgi Apparatus
Phase Separation
Biology Model Results Future Work
Phase Separation with Curvature
Snapshots in time of curvature induced phase separation
(a)
(b)
0.8
0.7
0.6
0.5
θs
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
x
Ross Magi
Curvature Induced Phase Separation in the Golgi Apparatus
Phase Separation
Biology Model Results Future Work
Other Shapes
Final shape depends on the initial perturbation
(a)
(b)
0.55
0.5
0.45
0.4
0.35
0.3
(c)
(d)
0.25
0.2
0.15
0.1
0.05
(a) 0.5(1 + 0.1 cos(6πx)) (b) 0.5(1 + 0.1 cos(8πx)) (c) 0.5(1 + 0.1 cos(10πx)) (d)
0.5(1 + 0.1 cos(12πx))
Ross Magi
Curvature Induced Phase Separation in the Golgi Apparatus
Phase Separation
Biology Model Results Future Work
Phase Separation by Imposed Curvature
Cisternal shape likely stabilized by the cytoskeleton
Snapshots in time of phase separation caused by imposed
shape
0.8
(a)
(b)
0.7
0.6
0.5
(c)
(d)
0.4
0.3
0.2
Ross Magi
Curvature Induced Phase Separation in the Golgi Apparatus
Phase Separation
Biology Model Results Future Work
Conclusions
Described a two phase fluid model where two phases were two
types of lipids
Incorporated shape into the equations describing fluid motion
Model demonstrated phase separation
Model produced shapes reminiscent of Golgi cisternae
Ross Magi
Curvature Induced Phase Separation in the Golgi Apparatus
Phase Separation
Biology Model Results Future Work
2D Phase Separation
A membrane is a 2D surface embedded in 3D
Many vesicle models have line tension as a crucial aspect of
the model
Line tension is a phenomenological energy penalty for the
interface between two phases
Two-phase model in 2D could exhibit similar membrane
deformations to models with line tension
Ross Magi
Curvature Induced Phase Separation in the Golgi Apparatus
Phase Separation
Biology Model Results Future Work
Thank You! Questions?
James P. Keener (Advisor)
Fellow U of U mathbio students
Funding
IGERT
RTG
U of U math department
Dr. Keener’s research grant
WWU Math Department!
Ross Magi
Curvature Induced Phase Separation in the Golgi Apparatus