Modeling vesicle traffic reveals unexpected consequences for Cdc42p-mediated polarity establishment.

Abstract

Polarization in yeast has been proposed to involve a positive feedback loop whereby the polarity regulator Cdc42p orients actin cables, which deliver vesicles carrying Cdc42p to the polarization site. Previous mathematical models treating Cdc42p traffic as a membrane-free flux suggested that directed traffic would polarize Cdc42p, but it remained unclear whether Cdc42p would become polarized without the membrane-free simplifying assumption.
We present mathematical models that explicitly consider stochastic vesicle traffic via exocytosis and endocytosis, providing several new insights. Our findings suggest that endocytic cargo influences the timing of vesicle internalization in yeast. Moreover, our models provide quantitative support for the view that integral membrane cargo proteins would become polarized by directed vesicle traffic given the experimentally determined rates of vesicle traffic and diffusion. However, such traffic cannot effectively polarize the more rapidly diffusing Cdc42p in the model without making additional assumptions that seem implausible and lack experimental support.
Our findings suggest that actin-directed vesicle traffic would perturb, rather than reinforce, polarization in yeast.

Full text

Page 1

Current Biology 21, 1–11, February 8, 2011 ª2011 Elsevier Ltd All rights reservedDOI 10.1016/j.cub.2011.01.012
Article
Modeling Vesicle Traffic Reveals
Unexpected Consequences for
Cdc42p-Mediated Polarity Establishment
Anita T. Layton,1Natasha S. Savage,2Audrey S. Howell,2
Susheela Y. Carroll,3David G. Drubin,3and Daniel J. Lew2,*
1Department of Mathematics, Duke University, Durham, NC
27708, USA
2Department of Pharmacology and Cancer Biology, Duke
University Medical Center, Durham, NC 27710, USA
3Department of Molecular and Cell Biology,
University of California, Berkeley, Berkeley, CA 94720, USA
Summary
Background: Polarization in yeast has been proposed to
involveapositivefeedbackloopwherebythepolarityregulator
Cdc42p orients actin cables, which deliver vesicles carrying
Cdc42ptothepolarizationsite.Previousmathematicalmodels
treating Cdc42p traffic as a membrane-free flux suggested
that directed traffic would polarize Cdc42p, but it remained
unclear whether Cdc42p would become polarized without
the membrane-free simplifying assumption.
Results: We present mathematical models that explicitly
consider stochastic vesicle traffic via exocytosis and endocy-
tosis,providingseveralnewinsights.Ourfindingssuggestthat
endocytic cargo influences the timing of vesicle internalization
in yeast. Moreover, our models provide quantitative support
for the view that integral membrane cargo proteins would
become polarized by directed vesicle traffic given the experi-
mentally determined rates of vesicle traffic and diffusion.
However, such traffic cannot effectively polarize the more
rapidly diffusing Cdc42p in the model without making addi-
tional assumptions that seem implausible and lack experi-
mental support.
Conclusions: Our findings suggest that actin-directed vesicle
traffic would perturb, rather than reinforce, polarization in
yeast.
Introduction
Polarity establishment and maintenance are crucial to the
function of many cell types. These processes are perhaps
best understood in the budding yeast Saccharomyces
cerevisiae, where genetic approaches identified a suite of
polarity regulators centered on the conserved GTPase
Cdc42p [1, 2]. Two positive feedback loops are thought to
contribute to Cdc42p polarization in yeast. One is a Turing-
type reaction-diffusion mechanism that can concentrate
GTP-Cdc42p in a cytoskeleton-independent manner [3–5].
The other is a vesicle-recycling feedback loop whereby
Cdc42p orients actin cables, which in turn deliver Cdc42p as
cargo on secretory vesicles [6–8]. Here we address the
requirements for such actin-mediated polarization.
Theactin-mediatedpositivefeedbackhypothesis(Figure1A)
was first proposed to explain the spontaneous polarization of
a GTP-locked mutant, Cdc42pQ61L, when it was overex-
pressed in G1-arrested yeast cells [6]. This mutant cannot
employ the Turing mechanism, and Cdc42pQ61Lpolarization
is blocked by treatments that disrupt actin and/or exocytosis,
consistent with the feedback hypothesis. However, disrupting
actin or exocytosis would have many indirect consequences
for cell physiology, potentially interfering with polarization
even if the proposed actin-mediated feedback were not
involved.
For actin-mediated feedback to operate, directed traffic of
Cdc42p on vesicles must suffice to concentrate Cdc42p at
the polarization site. A mathematical model incorporating
polarized delivery, diffusion, and endocytic retrieval of
Cdc42p (Figure 1B) indicated that once actin cables are polar-
ized, traffic of Cdc42p should indeed generate and maintain
a polarized distribution of Cdc42p at the plasma membrane
[7, 9]. However, this model assumed that Cdc42p trafficking
between internal pools and the plasma membrane can be
treated as a simple protein flux, without considering the
membranes that actually mediate the traffic. Using more real-
istic models with explicit vesicle traffic between the plasma
membrane and an internal compartment, we now show that
polarized vesicle delivery is incapable of maintaining a polar-
ized distribution of cargo unless the cargo diffuses very slowly
and is selectively concentrated into endocytic vesicles.
Because Cdc42p is not thought to fit these criteria, our find-
ings indicate a need to reevaluate the Cdc42p-actin feedback
hypothesis.
Results
Assumptions of the Cdc42p Trafficking Model
The previous model considered the polarized distribution
of Cdc42p in the plasma membrane to arise from delivery of
Cdc42p to a specified window and subsequent retrieval of
the protein by endocytosis before it diffused too far away
(Figure 1B). The founding assumptions include:
(1) All Cdc42p delivery is directed to a small patch
(or ‘‘window’’) in the plasma membrane.
(2) Endocytosisis moreactive
(rate constant m) than it is outside the window (rate
constant n).
withinthewindow
These assumptions are reasonable because in a polarized
cell, all detectable actin cables appear to be directed toward
the polarization site, and the actin patches that mark sites of
endocytosis are also concentrated near the polarization site
[10, 11].
(3) Cdc42p exocytosis occurs at a rate proportional to the
amount of Cdc42p in the internal pool.
(4) Cdc42p endocytosis from a given spot on the plasma
membrane occurs at a rate proportional to the concen-
tration of Cdc42p at that spot.
These are equivalent to assuming that Cdc42p is neither
concentrated in nor excluded from exocytic and endocytic
vesicles, so its rate of traffic is simply proportional to
its concentration (surface density) at the donor membrane.
*Correspondence: daniel.lew@duke.edu
Please cite this article in press as: Layton et al., Modeling Vesicle Traffic Reveals Unexpected Consequences for Cdc42p-Mediated
Polarity Establishment, Current Biology (2011), doi:10.1016/j.cub.2011.01.012

Page 2

This is reasonable given that no vesicular concentration or
exclusion mechanisms are currently known for Cdc42p.
A potentially problematic issue is the question of how much
of the intracellular Cdc42p pool is accessible for vesicular
traffic. A prominent subpool of Cdc42pQ61Lresides on vacuole
membranes, and a prominent subpool of endogenous Cdc42p
is cytoplasmic (GDI-bound); both seem unlikely to be a direct
source for exocytic Cdc42p. However, the assumptions may
bevalidifinsteadoftreating theentire internal poolas adonor,
we consider only a subpool of ‘‘transport-competent’’ Cdc42p
in recycling endosomal and trans-Golgi membranes. The
quantitative contribution of this pool to the total internal
Cdc42p is unknown.
(5) The polarized cell is at steady state: Cdc42p delivery by
exocytosis equals Cdc42p retrieval from the plasma
membrane by endocytosis.
The steady-state assumption also applies to total plasma
membrane area, which was assumed to be invariant.
Because the cells are growing (implying net increase in the
amounts of Cdc42p and membrane), this is clearly inaccurate
over long periods. However, it is likely to apply over the short
term.
Plasma Membrane
(Cdc42p diffuses)
Window
m
n
Internal Compartment
(Cdc42p well-mixed)
C
Internal Compartment
Exocytosis
Endocytosis
h
n
m
Window
B
Cdc42p
Vesicle
Actin cable
A
DE
GFP-Cdc42p
044668822
circumference (μm)
100
25
75
50
0
GFP intensity
Figure 1. Models for Concentrating Cdc42p
(A)Left:Cdc42pcanattachandorientactincables,which
delivervesiclescarrying
membrane. Right: the additional Cdc42p delivered by
vesicles orients more actin cables, which deliver more
vesicles with Cdc42p in a positive feedback loop.
(B) Previous mathematical
membrane-free Cdc42p fluxes to (rate constant h) and
from (rate constants m or n) a 1D perimeter, which by
rotationalsymmetryrepresents
membrane. All exocytic protein flux was directed to
a polarization window, and endocytosis was more active
in the window (m > n).
(C) We modeled Cdc42p fluxes as stochastic transfers of
vesicles (red, large exocytic vesicles; blue, small endo-
cytic vesicles) carrying Cdc42p between 2D plasma
membrane and internal compartments. m and n now
represent relative probabilities (per area) that endocytic
vesicles form at different locations.
(D) Polarized distribution of GFP-Cdc42p in yeast. Scale
bar represents 2 mm.
(E) Line scans of GFP-Cdc42p in the plasma membrane
of five polarized unbudded cells.
Cdc42pto theplasma
modelsconsidered
the2Dplasma
(6) Because the amount of membrane
delivered by exocytosis equals that
internalized by endocytosis, membrane
traffic can be ignored and Cdc42p traffic
can be treated as a direct protein flux to
and from the plasma membrane.
This assumption yields asubstantial simplifi-
cation on the mathematical side: it allows one
to describe stochastic delivery of vesicles
(discretemembrane
Cdc42p as a continuous simple flux. Consider,
however,theeffectofeliminatingassumption6
while retaining the others.
If Cdc42p traffics on vesicles, then it must be
present at some specific concentration on
those vesicles. Assumption 4 implies that the
Cdc42p concentration on endocytic vesicles
is simply that of the Cdc42p at the source plasma membrane:
we infer that the average Cdc42p concentration on endocytic
vesicles will be intermediate between the maximum (peak)
and minimum concentrations at the membrane. Assumption
5 (steady state) implies that the rate of exocytosis must equal
the rate of endocytosis for both Cdc42p and membrane: we
infer that the Cdc42p concentration on exocytic vesicles
must equal the average Cdc42p concentration on endocytic
vesicles. Therefore, the Cdc42p concentration on exocytic
vesicles must be lower than the peak Cdc42p concentration
at the plasma membrane, and delivery of new vesicles to
that site would dilute the local Cdc42p rather than concentrate
it. But then the peak concentration cannot be maintained at
steady state. This argument suggests that without the simpli-
fying assumption of membrane-free Cdc42p flux, the system
would not in fact maintain a polarized steady state.
packets)carrying
Explicit Modeling of Cdc42p Traffic by Exocytosis
and Endocytosis
We set out to devise a model that incorporates the vesicular
carriers. To model Cdc42p flux, we need to know the rate at
which vesicles traffic in each direction, the spatial distribution
of vesicle fission and fusion events, the concentration of
Current Biology Vol 21 No 3
2
Please cite this article in press as: Layton et al., Modeling Vesicle Traffic Reveals Unexpected Consequences for Cdc42p-Mediated
Polarity Establishment, Current Biology (2011), doi:10.1016/j.cub.2011.01.012

Page 3

Cdc42p on the vesicles, and the membrane area of the
vesicles. Some of this information is available in the literature;
for the rest, we initially employed assumptions 1–5 above
(Table 1). Vesicle traffic was modeled as the stochastic trans-
fer of membrane packets carrying Cdc42p between a well-
mixedinternalcompartment(representingtheendomembrane
system relevant to Cdc42p recycling) and the plasma
membrane (Figure 1C). At the plasma membrane, traffic was
directed to a central window, and Cdc42p distribution evolved
as a result of both membrane traffic and diffusion. We call this
the ‘‘bulk traffic’’ model, to indicate that vesicular Cdc42p
concentration (surface density) is simply the concentration at
thedonormembranethatgaverisetothevesicle,andtodistin-
guish it from subsequent models.
The Uniform Cdc42p Distribution Represents a Stable
Steady State
A simulation initiated with equal Cdc42p concentration in the
internal compartment and all over the plasma membrane
remained unpolarized despite having all exocytosis directed
to the window (Figure 2B). Because all vesicles carry the
same concentration of Cdc42p as that on the plasma
membrane, vesicle fusion and fission does not change the
local Cdc42p concentration. Thus, the unpolarized state is
stable despite highly polarized vesicle trafficking.
The stability of the unpolarized state in our model stands in
markedcontrasttothepredictionsofthepreviousmodel[7,9].
In effect, that model had Cdc42p trafficking at infinite concen-
tration, so that the local concentration of Cdc42p at the target
membrane was always increased by the flux. By incorporating
the vesicular carriers, our model allows for traffic to have no
effect, or even to dilute Cdc42p, depending on the relation
between the Cdc42p concentrations in the donor and target
membranes.
The Polarized Cdc42p State Is Unstable and Decays
to the Uniform Steady State
We next initiated simulations with Cdc42p at 10-fold higher
concentrationwithinthewindowthanelsewhere(adistribution
roughly similar to that in a polarized yeast cell; Figure 2A). This
polarized distribution rapidly decayed toa uniform distribution
(Figures 2C and 2D). Both exocytic and endocytic events
collaborate with diffusion to dissipate polarity: endocytosis re-
moves Cdc42p from the window, and fusion of exocytic vesi-
cles dilutes Cdc42p in the window. The dissipative effect of
vesicle traffic is small compared with that of the diffusion coef-
ficient estimated for Cdc42p (D = 0.036 mm2/s; Figure 2E) [7]
but quite large compared with that of the diffusion coefficient
estimated for integral membrane proteins (D = 0.0025 mm2/s;
Figure 2F) [12]. Thus, a model incorporating vesicle traffic
and adhering to assumptions 1–5 above cannot maintain
a polarized state.
What If Cdc42p Were to Be Concentrated into Exocytic
Vesicles?
In order to develop polarity in our model, a key requirement is
thattheconcentrationofCdc42pinexocyticvesiclesbehigher
than that at the peak of the Cdc42p distribution in the window.
For as long as this holds true, each exocytic event should, at
least transiently, increase polarity. Because the GFP-Cdc42p
distribution does not reveal any internal compartment
harboring Cdc42p at a concentration comparable to that in
the window (Figure 1D), satisfying this condition requires
that Cdc42p be more concentrated in the exocytic vesicles
than it is in the internal compartment from which the vesicles
emerge. Although not known to occur for Cdc42p, many
integral membrane proteins do become more concentrated
in vesicles compared to donor membranes (up to 10-fold, as
estimated from electron microscopy of an abundant cargo
protein targeted to the plasma membrane [13]). We next
performed simulations in which Cdc42p was concentrated
10-fold during exocytic vesicle formation.
Starting from a homogeneous state, this new model did
initially develop weakpolarity (Figure 2G; first2 min). However,
the concentrated Cdc42p delivered by each exocytic vesicle
dissipated rapidly, preventing robust polarization. Either
increasing the vesicle trafficking rate (Figure 2H) or decreasing
the diffusion constant (Figure 2I) enabled development of
stronger polarity, implying that the inability to polarize effec-
tively was due to an imbalance between slow vesicle traf-
ficking and rapid diffusion.
On longer timescales (Figure 2J; 30 min), the polarizing
effect of exocytosis was eliminated, resulting in a uniform
distribution of Cdc42p. Although faster vesicle trafficking
improved the initial polarization, polarity then dissipated
rapidly (Figure 2K). Slowing diffusion was more effective (Fig-
ure2L).ThelossofpolaritywasduetoanettransferofCdc42p
from the internal compartment to the plasma membrane (Fig-
ure 2M). As Cdc42p became depleted from the internal
compartment,exocyticvesicles carried lessand less
Table 1. Model Parameters
ParameterValueCommentsReference
Plasma membrane area
Polarization window area
Exocytic vesicle area
Endocytic vesicle area
Frequency of exocytosis
Frequency of endocytosis
Ratio of endocytosis probabilities
Area of internal compartment
Cdc42p diffusion constant
v-SNARE diffusion constant
Endocytic patch cargo trapping time t
Endocytic patch fill level
Maximum cargo-trapping time
Septin ring inner diameter
Septin ring outer diameter
78.5 mm2
0.785 mm2
0.0314 mm2
0.00785 mm2
0.42 vesicles/s
1.67 vesicles/s
40
55–78.5 mm2
0.036 mm2/s
0.0025 mm2/s
8 s
10
24 s
1 mm
1.6 mm
5 mm diameter sphere
1 mm diameter circle
0.1 mm diameter sphere
0.05 mm diameter sphere
based on steady-state assumption
25 actin patches/cell and 15 s actin patch lifetime
probability/bin inside polarization window higher than outside
13 plasma membrane in bulk traffic model, 0.73 in sink models
diploid unbudded cell
[7]
[36]
[37]
[7]
[30, 38]
[7]
see text
[7]
[12]
see text
see text
see text
[45]
[45]
assumed equal for pheromone receptor
for uniform-time model
for uniform-fill model
for uniform-fill model
surrounds polarization window
Vesicle Trafficking and Polarity
3
Please cite this article in press as: Layton et al., Modeling Vesicle Traffic Reveals Unexpected Consequences for Cdc42p-Mediated
Polarity Establishment, Current Biology (2011), doi:10.1016/j.cub.2011.01.012

Page 4

5
0
120060
Time (s)
5
4
3
2
1
0
A
5
4
3
2
1
0
5
4
3
2
1
0
1 s120 s
B
5
4
3
2
1
0
120 s1 s
5
4
3
2
1
0
C
Slower Diffusion:
5
0
5
0
5
0
2x
4x
8x
1200 60
Time (s)
I
5
0
5
0
5
0
2x
4x
8x
Faster Traffic:
120060
Time (s)
H
G
10x [Cdc42p] in exocytic vesicles:
5
0
120060
Time (s)
D
Kymograph:
J
K
L
8x Slower Diffusion:
M
8x Faster Traffic:
1
4
7
10
30
0 2010
Time (min)
Peak-to-trough ratio:
1
4
7
10
30
02010
Time (min)
Peak-to-trough ratio:
1
4
7
10
30
02010
Time (min)
Peak-to-trough ratio:
5
0
30020 10
Time (min)
5
0
30 0 20 10
Time (min)
5
0
300 20 10
Time (min)
membrane
internal
Time (min)
[Cdc42p]
0
5 10 1520 2530
0
1
2
8x Faster Traffic:
membrane
internal
Time (min)
[Cdc42p]
0
5 1015 20 25 30
0
1
2
[Cdc42p] on membrane and internal compartment:
1
4
7
10
600 20 40
Time (s)
80
100
Peak-to-trough ratioPeak-to-trough ratio
1
4
7
10
6024
Time (s)
8
10
D=0.036 μm2/sD=0.0025 μm2/s
diffusion
diffusion + vesicles
diffusion
diffusion + vesicles
EF
Figure 2. Directed Traffic of Cdc42p Cannot Sustain Polarization, Even If Cdc42p Is Concentrated into Exocytic Vesicles
(A) Cdc42p distribution from one of the cells in Figure 1E is displayed on a square plasma membrane by assuming rotational symmetry. Color bar indicates
Cdc42p concentration in arbitrary units.
(B) The uniform Cdc42p distribution is stable. Left: Cdc42p distribution at t = 1 s. Right: Cdc42p distribution at t = 120 s.
(C) The polarized Cdc42p distribution is unstable. Left: Cdc42p distribution at t = 1 s. Right: Cdc42p distribution at t = 120 s.
(D) Kymograph of the simulation from (C) showing a slice through the middle of the plasma membrane (y axis) as time progresses (x axis).
(E) Plot of the peak-to-trough ratio of Cdc42p concentrations in the 2D plasma membrane from simulations performed as in (C). Red line indicates dissipa-
tionofthestartingpolaritybydiffusionalone.Bluelineindicatesaverageof20simulations includingvesicletrafficaswellasdiffusion.Thisdissipation rateis
comparable to that of diffusion alone with D = 0.04 mm2/s.
(F) As in (E),except that protein diffusion was modeled tobe that of integralmembrane proteins (D = 0.0025 mm2/s). Here the dissipation with vesicle trafficis
comparable to that of diffusion alone with D = 0.008 mm2/s.
(G) Kymograph of a 120 s simulation in which Cdc42p is concentrated 10-fold from the internal compartment into exocytic vesicles.
(H) Simulations as in (G), but with 2-, 4-, or 8-fold faster vesicle traffic.
Current Biology Vol 21 No 3
4
Please cite this article in press as: Layton et al., Modeling Vesicle Traffic Reveals Unexpected Consequences for Cdc42p-Mediated
Polarity Establishment, Current Biology (2011), doi:10.1016/j.cub.2011.01.012

Page 5

Cdc42p, reducing their polarizing effect. Meanwhile, Cdc42p
accumulation at the plasma membrane made the gradient
shallower. Faster vesicle trafficking accelerated the transfer
of Cdc42p (Figure 2M), speeding the loss of polarity.
These simulations indicate that even if we alter the bulk
traffic scenario to include a step that concentrates Cdc42p
10-fold during exocytic vesicle biogenesis, trafficking does
not lead to a strong or sustained polarized state, for two
reasons. First, there is a kinetic mismatch between the rate
of vesicle traffic and the speed of Cdc42p diffusion, so that
the polarizing effect of one vesicle is largely dissipated before
the next vesicle arrives. Second, because Cdc42p is concen-
trated into exocytic but not endocytic vesicles, there is a net
transfer of Cdc42p from the internal compartment to the
plasma membrane. Once the relative amounts of Cdc42p in
the two compartments reach steady state (with 10 times
more Cdc42p on the plasma membrane than in the internal
compartment), exocytic vesicles (though still 10-fold concen-
trated relative to what is left in the internal compartment) no
longer carry enough Cdc42p to increase its concentration in
the window. The situation is then equivalent to that of the
bulk traffic model: all vesicles carry the same concentration
of Cdc42p as that on the plasma membrane, and a polarized
Cdc42p distribution rapidly decays to a uniform steady state.
What If Cdc42p Were to Be Concentrated into Endocytic
Vesicles?
Could the problem discussed above be circumvented if
Cdc42p were actively concentrated into endocytic vesicles
as well as exocytic vesicles? Although not known to occur
for Cdc42p, concentration of cargo in endocytic vesicles is
welldocumentedformanyintegralmembraneproteins,thanks
to a variety of ‘‘endocytosis signals’’ present in their cyto-
plasmic tails [14, 15]. Of particular interest are the vesicle-
soluble NSF attachment protein receptors (v-SNAREs) that
target fusion of exocytic vesicles with the plasma membrane.
v-SNAREs are concentrated into both exocytic and endocytic
vesicles, and in yeast, v-SNAREs exhibit a polarized distribu-
tion in the plasma membrane. Active endocytosis has been
demonstrated to be essential for the polarized distribution of
v-SNAREs, and appending a heterologous endocytosis signal
toanotherwiseunpolarizedintegral membraneproteincauses
it to become polarized [12]. Thus, we speculated that concen-
trating Cdc42p into endocytic (as well as exocytic) vesicles
might allow the model to sustain a polarized steady state.
We next develop such a model.
Modeling the Concentration of Endocytic Cargo Proteins
into Patches
Because Cdc42p in the plasma membrane is not homoge-
neously distributed, the amount of Cdc42p present on a given
endocytic vesicle will depend on the local Cdc42p concentra-
tion profile and the effectiveness with which Cdc42p becomes
trapped in the forming vesicle. To develop a realistic model for
thiscomplexprocess,webeganbyconsidering theknowncell
biology of endocytosis in yeast and deriving realistic parame-
ters from well-characterized endocytic cargo.
The first step in vesicle biogenesis is the assembly of a coat
containing clathrin and adaptor proteins at a patch on the
plasma membrane. Accumulation of cargo into this patch
then occurs via interactions between the specific endocytosis
signals on the cargo and binding sites for those signals on
adaptor proteins [15, 16]. After a variable time interval
(w30–210 s [17]), actin polymerization is initiated at the patch,
and the membrane undergoes invagination and scission
(a process taking w15 s) to become an endocytic vesicle
(Figure 3A).
We adapted our model for endocytosis to include cargo-
trapping and internalization steps. For cargo trapping, we
assume that for a time interval t between patch formation
and internalization, the patch acts as a diffusion sink: cargo
can diffuse in but not out. For internalization, we introduce
a15s‘‘deadtime,’’duringwhichthetrappedcargoisinsulated
from the neighboring plasma membrane but has not yet been
transferred to the internal compartment. Simulations of diffu-
sion sink behavior starting from plasma membranes contain-
ing uniform concentrations of cargo indicated that the degree
of cargo accumulation into a patch is approximately linear
with t (Figure 3B) and diffusion constant D (Figure 3C) in the
relevant parameter range.
Modeling endocytic patch behavior also requires that we
specifyrulesgoverningtheswitchbetweenthecargo-trapping
phaseandtheinternalizationphaseofendocytosis:whatisthe
trigger for actin polymerization? The answer to this question is
not known. We considered two plausible hypotheses that led
to distinct modeling strategies. (1) The switch is governed by
a timer: formation of a patch starts a clock that promotes actin
polymerization and internalization after some fixed t (we call
this the ‘‘uniform-time’’ model). (2) The switch is governed by
patch cargo content: when this reaches a certain ‘‘fill level’’ f,
actin polymerization and internalization occur (we call this
the‘‘uniform-fill’’model).Wenotethatintheuniform-fillmodel,
the particular cargo under consideration stands in for the
aggregate of all cargo proteins trapped by the patch. More-
over, the model parameters t and f incorporate information
on several unknown factors including the concentrations of
adaptors, the efficiency of cargo trapping, and the patch life-
times. A concern for the uniform-fill model, highlighted by
preliminary simulations, is that patches forming in regions
devoid of cargo could have extremely long lifetimes. To avoid
such unphysiological effects, we included a provision that if
a patch had failed to fill up by a designated maximum t, it
would go ahead and internalize with whatever cargo was
present.
Parameter Estimation Based on Data from Unpolarized
Cells
To assess what values of t or f would accurately represent
endocytosis for a well-characterized cargo, we simulated
theinternalizationofpheromone/pheromone
complexes in unpolarized cells. With a diffusion constant
receptor
(I) Simulations as in (G), but with 2-, 4-, or 8-fold slower diffusion.
(J) Left: kymograph of a 30 min simulation started as in (G). Right: plot of the peak-to-trough ratio of Cdc42p concentrations in the 2D plasma membrane for
the same simulation. Inset: first 2 min, showing that each vesicle fusion event yields a peak that dissipates rapidly.
(K) Simulation as in (J), but with 8-fold faster vesicle traffic.
(L) Simulation as in (J), but with 8-fold slower diffusion.
(M) Plot of average Cdc42p concentration on the plasma membrane(red line) and in the internal compartment (blue line) for the simulations shown in (J) (left)
and (K) (right). Vesicle traffic mediates a net transfer of Cdc42p from the internal compartment to the plasma membrane, equilibrating when the membrane
[Cdc42p] is 10 times the internal [Cdc42p].
Vesicle Trafficking and Polarity
5
Please cite this article in press as: Layton et al., Modeling Vesicle Traffic Reveals Unexpected Consequences for Cdc42p-Mediated
Polarity Establishment, Current Biology (2011), doi:10.1016/j.cub.2011.01.012

Page 6

for integral membrane proteins of 0.0025 mm2/s [12], simula-
tions yielded the kinetics of pheromone/receptor internaliza-
tion shown in Figure 3D. Experimental determinations
suggest t1/2values of 6–13 min [18–22], corresponding to t
values of 6–16 s (Figure 3D), based on which we selected
a value of t = 8 s. With this t, integral membrane endocytic
cargo proteins would be concentrated w6-fold into the
patch (Figures 3B and 3C). Comparable pheromone internal-
ization kinetics are predicted by the uniform-fill model with
f = 10.
Cargo recycled between the plasma membrane and the
internal compartment would, in unpolarized cells, reach
a steady-state distribution reflecting the relative degree to
which it gets concentrated into exocytic and endocytic vesi-
cles. If cargo is concentrated 10-fold in exocytic vesicles and
6-fold in endocytic vesicles, a recycling protein would reach
a steady state in which the cargo concentration in the internal
compartment was 60% of that on the plasma membrane.
This is in reasonable agreement with the visual impression
from images of either GFP-Cdc42p (Figure 1D) or v-SNARE
[23, 24] distribution in yeast.
Polarized Traffic of a v-SNARE Yields a Polarized Steady
State
Toassesswhether polarizedtrafficwouldgenerateapolarized
v-SNARE distribution in our model, we conducted simulations
with 10-fold concentration of cargo into exocytic vesicles and
with endocytosis occurring using either the uniform-time
model with t = 8 s (Figure 4A) or the uniform-fill model with
f = 10 (Figure 4B). Vesicle trafficking frequencies and spatial
distributions were as in the bulk traffic model. We started
with the uniform steady state resulting from simulated traffic
in unpolarized cells and switched to polarized traffic. Both
models rapidly generated a highly polarized state with a broad
peak in protein concentration that was maintained for >1 hr.
Thus, unlike a bulk cargo, a protein with the trafficking charac-
teristics of a v-SNARE would become polarized by directed
traffic.
Actin
coat
cargo
1. Patch formation
2. Cargo trapping
3. Internalization
C
AB
D
81012 1416 18
2
4
6
8
10
0
0
cargo trapping time τ (s)
Fold cargo concentration
for D = 0.0025 μm2/s
100
10
1
0.020.002 0.0002
Diffusion constant (μm2/s)
Fold cargo concentration
0
1020 304050 60
Time (min)
0
20
40
60
% remaining on surface
80
100
2
4
8
16
32
64
τ(s)
for τ = 8 s
642
Figure 3. Simulating Endocytic Patches as Diffusion
Sinks
(A) Schematic of endocytosis.
(B) Simulations of cargo accumulation by diffusion sinks
in unpolarized cells were used to assess the degree to
which patches would concentrate a cargo with D =
0.0025 mm2/s in a given trapping time t.
(C) Simulations as in (B) were used to assess the degree
to which patches would accumulate a cargo with varying
diffusion constant in t = 8 s.
(D) Simulations of pheromone/receptor endocytosis
assuming theindicated cargo trapping times (t)forendo-
cytic patches. The gray lines indicate that 50% of the
cargo would be internalized in 11 min assuming t = 8.
Behavior of Endocytic Patch Proteins in
Polarized Cells Suggests that Cargo
Triggers Internalization
Although both uniform-time and uniform-fill
models yielded a polarized v-SNARE distribu-
tion, they differed in significant ways. With the
uniform-time model, patches that formed
in the window accumulated much more
v-SNARE than those that formed at distant
sites, so the amount of v-SNARE internalized
per patch was heterogeneous (Figure 4C). With the uniform-
fill model, patches that formed in the window filled up much
faster than those at distant sites, so patch t was heteroge-
neous (Figure 4D). The two models yielded very different
spatial distributions of patches engaged in cargo trapping
(green in Figures 4E and 4F) versus internalization (red in
Figures 4E and 4F). With the uniform-time model, the green/
red patch ratio would be similar everywhere (Figure 4E),
whereas with the uniform-fill model, the ratio would be signif-
icantly smaller near the window, where patches fill faster
(Figure 4F).
Todistinguishbetweenthesemodels,weimagedyeastcells
containing Ede1p-GFP, a marker of the cargo-trapping phase
of endocytosis, and Abp1p-RFP, a marker of the internaliza-
tion phase of endocytosis. In budded cells, all exocytic traffic
is directed to the bud (analogous to the central window in the
simulations), and the distribution of marker proteins dramati-
cally supported the uniform-fill model: the Ede1p-GFP/
Abp1p-RFP ratio was much smaller in buds than in mothers
(Figure 4G). Direct examination of patch lifetimes showed
that whereas Abp1p patches displayed uniform 13–14 s life-
times, Ede1p patches were longer lived in mothers than in
buds of polarized cells (Figure 4H, upper inset). Measurement
of patch lifetimes requires that the entire patch lifetime be
included in a movie, so the data used to calculate these aver-
ages excluded many Ede1p patches that did not internalize by
the end of our 4 min movies and therefore underestimate the
true Ede1p patch lifetime in mother cells. Indeed, we observed
a significant number of Ede1p patches in polarized mother
cells that persisted for the entire duration of the movies: inclu-
sion of these patches yielded the lifetime distributions shown
in Figure 4H (top). Once cells progressed to the mature bud
stage where secretion of cargo was unpolarized, Ede1p patch
lifetimes were similar in mothers and buds, and very long-lived
patches were no longer detected (Figure 4H, bottom). We
conclude that the uniform-fill model better represents the situ-
ation in yeast, suggesting that cargo filling can trigger endo-
cytic patch internalization.
Current Biology Vol 21 No 3
6
Please cite this article in press as: Layton et al., Modeling Vesicle Traffic Reveals Unexpected Consequences for Cdc42p-Mediated
Polarity Establishment, Current Biology (2011), doi:10.1016/j.cub.2011.01.012

Page 7

Septin-Mediated Targeting of Endocytosis Sculpts
the Polarized Protein Distribution
Figure 4G also confirms previous studies indicating that
patches tend to cluster at the mother-bud neck. Because of
their close packing, it has not been possible to confirm that
the patches at the neck behave in the same stereotyped
manner as those elsewhere, but here we assume that they
do. Neck localization is due to the influence of the septins,
a family of cytoskeletal polymers that assemble in a ring
surrounding the polarization site and at the neck [17, 25, 26].
Unbudded but polarized cells (Figure 4I) also displayed
patches clustered in a ring (rather than uniformly within the
window as assumed in previous models), consistent with
a major influence of septins on patch placement. To incorpo-
rate these findings, we changed the spatial probability distri-
bution for endocytosis in the model to bias patch formation
0
3000
0
# patches
80 206040
cargo
0
3000
0
# patches
24
8
16
τ (s)
A
BF
C
5
0
Time (min)
2040600
Uniform patch lifetime
5
0
Time (min)
Ede1p-GFP
20 40600
Uniform cargo content
D
E
G
Abp1p-mCherryMerge
H
I
J
5
0
5
0
5
0
5
0
5
0
30%
40%
50%
60%
70%
5
Time (min)
10 150
Septin-biasedendocytosis
100
75
50
25
0
% Ede1p patches > τ
100
75
50
25
0
Buds
Mothers
MB
0
140
% Ede1p patches > τ
Polarized
Cells
Mothers
MB
τ (s)
0 240180120 60
0
Unpolarized
Cells
140
Buds
τ (s)
0240180 12060
Figure 4. Polarization of v-SNAREs by Vesicle Traffic
Simulations were performed using uniform-time
(A, C, and E) or uniform-fill (B, D, and F) models.
(A and B) Starting from the unpolarized steady state,
modelcellswereswitchedtopolarizedtrafficat1min
and simulated for 1 hr. Kymographs show that both
models developed and sustained a polarized cargo
distribution.
(C) Histogram showing amount of cargo internalized
per patch in a uniform-time simulation.
(D) Histogram showing patch t in a uniform-fill simu-
lation.
(E and F) Snapshots of endocytic patch distributions
in uniform-time (E) or uniform-fill (F) simulations.
Green indicates patches during cargo-trapping
phase; red indicates patches during internalization
phase.
(G and I) Micrographs of cells harboring the patch
markers Ede1p-GFP (green) and Abp1p-mCherry
(red).
(G) Budded cells show more red than green patches
in the bud but more green than red patches in the
mother. Scale bar represents 2 mm.
(H) Ede1p and Abp1p patch lifetimes in mothers and
buds. Main graphs: Ede1p-GFP lifetimes. x axis
shows time t; y axis shows percent of patches with
lifetimeslongerthant.Datawerebinnedin10sincre-
ments. n=40–55patchesineach dataset, from4min
movies. Polarized (upper panel) and unpolarized
(bottom panel) cells were distinguished by moni-
toring Abp1p-RFP distribution. Allcellsweremedium
to large budded. Inset: average (6 standard devia-
tion) patch lifetimes for Ede1p-GFP (green) and
Abp1p-RFP (red) in buds (B) and mothers (M).
(I) Unbudded polarized cells show a ring of patches
surrounding the polarization site. Scale bar repre-
sents 1 mm.
(J) Simulations as in (B) (uniform cargo content), but
withendocytosis biased towardthe septinring rather
than the central window. Left: kymographs of cargo
distribution. Right: snapshots of endocytic patch
distributions. Each row is a different simulation
assuming that the indicated fraction (30%–70%) of
patches formed within the ring. The ring has inner
diameter 1 mm and outer diameter 1.6 mm and is
most clearly seen by looking at patch locations in
the high-bias (70%) simulations.
to a ring surrounding the window. Simula-
tions using the uniform-fill model in which
increasing numbers of patches formed
within the ring produced increasingly narrow polarized
v-SNARE distributions in the plasma membrane (Figure 4J).
Thus, septin-biased endocytosis has the potential to sculpt
the polarized distribution of recycling cargo proteins in the
plasma membrane.
Could Active Endocytosis Polarize Cdc42p?
Returning to the question of Cdc42p polarization, does the
successful polarization of a v-SNARE in our models (Figure 4)
imply that Cdc42p could be similarly polarized if it were also to
be concentrated into endocytic as well as exocytic vesicles?
The v-SNARE diffusion constant (0.0025 mm2/s) [12] is much
slower than that estimated for Cdc42p (0.036 mm2/s) [7].
To model Cdc42p behavior, we added a ‘‘passenger’’ cargo
to the v-SNARE uniform-fill model with 50% septin-biased
endocytosis. The passenger, Cdc42p, is carried by the same
Vesicle Trafficking and Polarity
7
Please cite this article in press as: Layton et al., Modeling Vesicle Traffic Reveals Unexpected Consequences for Cdc42p-Mediated
Polarity Establishment, Current Biology (2011), doi:10.1016/j.cub.2011.01.012

Page 8

vesicles as the v-SNARE but does not affect the timing of
vesicle internalization. This strategy allows independent
control of the degree to which Cdc42p becomes concentrated
into exocytic and endocytic vesicles. For the latter, we specify
a Cdc42p ‘‘fill level’’ fCdc42pthat reflects factors such as the
trapping efficiency and total capacity of endocytic patches
for Cdc42p.
Simulations with this ‘‘dual-cargo’’ model showed that if
Cdc42p were to be concentrated into exocytic and endocytic
vesicles to the same extent as v-SNAREs (f = 10), the rapid
diffusion of Cdc42p would dissipate the polarity created by
vesicle traffic (Figure 5A). Although a polarized distribution
could be produced by further increasing the vesicular
Cdc42p concentration (fCdc42p= 30–40), such polarization dis-
played dramatic fluctuations in Cdc42p distribution (Figures
5B–5D). Filming of GFP-Cdc42p [8] has not revealed the bright
spots or extreme fluctuations predicted by this model. Thus,
given the quantitative constraints on vesicle trafficking
frequencies and the published Cdc42p diffusion constant,
Cdc42p could not be effectively polarized via trafficking.
Discussion
Effect of Cdc42p Traffic on Polarization
Polarity establishment in yeast is thought to involve a positive
feedback loop in which Cdc42p is delivered to the polarization
site by vesicles traveling along actin cables, which themselves
are oriented by Cdc42p. Previous mathematical models
treated Cdc42p traffic as a direct protein flux, without taking
into consideration the membranes that actually carry the
Cdc42p. In such models, all traffic concentrates Cdc42p at
the target membrane. Realistically, however, vesicle fusion
would only concentrate Cdc42p if the Cdc42p concentration
on the vesicle were higher than that in the target membrane.
We present models of Cdc42p traffic that explicitly consider
the vesicular carriers and show that bulk traffic of Cdc42p
would dissipate, not enhance, polarity (Figure 2).
Our simulations suggest that in contrast to bulk cargo, an
integral membrane cargo protein that diffuses slowly and is
actively concentrated into both exocytic and endocytic vesi-
cles would become effectively polarized (Figure 4). The key
parameters in these models (vesicle dimensions and traf-
ficking frequencies, degree of cargo trapping into endocytic
patches, and diffusion constant) are all constrained within
a factor of w2 by experimental data, so the ability of the
models to reproduce the experimentally observed v-SNARE
distribution provides strong quantitative support for the pre-
vailing hypothesis that v-SNARE polarization is due to polar-
ized vesicle traffic [12].
In budded cells, v-SNAREs are concentrated in the bud and
largely absent from the mother plasma membrane [12]. One
possible basis for this is a diffusion barrier at the mother-bud
neck. Indeed, the septin ring, which localizes to the neck, is
thought to create such a barrier [27]. Recent studies sug-
gestedthattheseptinringmightalsopromoteendocyticpatch
formation [17, 25]. We modeled the effect of septin-biased
endocytosis on the distribution of cargo in the plasma
membrane and found that when vesicle delivery to a central
window is coupled with a sufficient septin bias of endocytosis
around the window, recycling cargo proteins are restricted to
the vicinity of the window. This effect provides an alternative
explanation for why v-SNAREs and similar cargo proteins are
Cdc42p
B 20x [Cdc42p] in exocytic vesicles
fCdc42p= 20 in endocytic vesicles
Cdc42p
C 30x [Cdc42p] in exocytic vesicles
fCdc42p= 30 in endocytic vesicles
Cdc42p
D 40x [Cdc42p] in exocytic vesicles
fCdc42p= 40 in endocytic vesicles
Cdc42p
10x [Cdc42p] in exocytic vesicles
fCdc42p= 10 in endocytic vesicles
A
0
5
0 1020 155
10 15
Time (min)
0
10
20
30
0
5
0 1020 155
1015
Time (min)
0
10
20
30
0
5
0 10 20 155
1015
Time (min)
0
10
20
30
0
5
0 10 20155
10 15
Time (min)
0
10
20
30
0
5
0 10 20155
1015
Time (min)
v-SNARE
0
10
20
30
0
5
0 1020155
10 15
Time (min)
v-SNARE
0
10
20
30
0
5
0 10 20 155
10 15
Time (min)
v-SNARE
0
10
20
30
0
5
0 1020 155
10 15
Time (min)
v-SNARE
0
10
20
30
Figure 5. Simulating Traffic of Vesicles Carrying Both v-SNAREs and Cdc42p
Simulations were performed assuming a uniform cargo content with 50% septin-biased patches as in Figure 4J. In addition to v-SNARE cargo, a more
rapidly diffusing ‘‘passenger’’ cargo representing Cdc42p traffics on the same vesicles. The fold concentration of Cdc42p into exocytic vesicles and the
Cdc42p filllevel fCdc42pwereset to10 (same as for v-SNARE) in(A), 20 in(B), 30in (C), and 40in (D). Upper panels: kymographs of Cdc42p distribution during
20 min simulation and plots showing the [Cdc42p] peak-to-trough ratio during an interval between 10 and 15 min. Lower panels: kymographs of v-SNARE
distribution for the same simulation and plots showing the [v-SNARE] peak-to-trough ratio in the same interval.
Current Biology Vol 21 No 3
8
Please cite this article in press as: Layton et al., Modeling Vesicle Traffic Reveals Unexpected Consequences for Cdc42p-Mediated
Polarity Establishment, Current Biology (2011), doi:10.1016/j.cub.2011.01.012

Page 9

restricted to the bud. Moreover, like septin-restricted diffu-
sion, it would also keep such proteins focused near the polar-
ization site in unbudded cells.
We recently generated a yeast strain that was synthetically
rewired by fusing a normally cytoplasmic polarity protein to
a v-SNARE [28]. The ability of this rewired strain, in which
actin-independent polarization was disabled, to develop
polarity suggests that actin-mediated vesicle traffic can
concentrate a polarity protein with the trafficking characteris-
tics of a v-SNARE to a sufficient degree to enable positive
feedback. Together, the rewired strain and the models devel-
oped here support the idea that an actin-mediated positive
feedback loop can operate successfully if a polarity protein
has suitable trafficking characteristics.
Do any endogenous polarity proteins exhibit these charac-
teristics? None of the known polarity proteins in yeast are inte-
gral membrane proteins, and there is as yet no evidence to
indicate that any can be actively endocytosed. Most polarity
regulators are cytoplasmic proteins that associate only tran-
siently with the plasma membrane, and although Cdc42p itself
is a peripheral membrane protein, it is thought to diffuse quite
rapidly relative to v-SNAREs. We show that vesicle traffic
would be ineffective in polarizing such rapidly diffusing
proteins even if they did become concentrated in forming
vesicles.
Cdc42p and other polarity regulators become polarized
even in the complete absence of polymerized actin in yeast
[3, 29], so it is clear that actin-independent polarization mech-
anisms exist. Actin-independent polarization is suggested to
operate via a positive feedback loop in which GTP-Cdc42p
recruits a rapidly diffusing cytoplasmic complex containing
the Cdc42p-directed GEF, which then promotes GTP loading
of neighboring Cdc42p [4]. This system acts via a Turing-
type reaction-diffusion mechanism [5] and in principle suffices
to account for polarity establishment in yeast.
New Insights from Modeling Endocytosis
Endocytic patches form at the plasma membrane, stay there
(presumably trapping cargo) for a time, and then internalize
in a stereotypical sequence [30]. We developed two models
for cargo endocytosis: one assumed that patch lifetime is
uniform, whereas the other assumed that patch cargo content
is uniform (i.e., patches are triggered to internalize once they
trap a certain amount of cargo). The models behaved similarly
in unpolarized cells, but they differed in significant ways in
polarized cells. In particular, if patches are triggered to inter-
nalizeoncetheytrapacertainamountofcargo,thenpatchlife-
times should be much shorter in the bud (where abundant
cargo rapidly fills the patch) than in the mother. Using markers
of the cargo-trapping and internalization phases of endocy-
tosis, we confirmed this prediction, suggesting that endocytic
cargo can promote internalization.
How would cargo trigger internalization? Internalization is
initiated by Arp2/3-mediated actin polymerization [31], and
some endocytic cargo adaptors can inhibit Arp2/3 [32, 33].
We speculate that cargo binding to the adaptors might relieve
their inhibition of Arp2/3, enabling internalization once suffi-
cient cargo binding occurs.
In mammalian cells, cargo is thought to bias clathrin patch
biogenesistoward productive
opposed to abortive disassembly [34, 35], but abortive events
have not been described in yeast. Why would cargo influence
patch internalization instead of initial maturation in yeast? One
possibility is suggested by the observation that membrane
coated-pitformation as
proteins diffuse much more slowly in the yeast plasma
membrane than in mammalian plasma membranes [12]. With
slow cargo diffusion, a longer time would be required for yeast
clathrin patches to assess the local cargo environment.
Regulation of a later step in the endocytic process may allow
sufficient time for accurate cargo monitoring, yielding a more
effectivestrategy tocouple
internalization.
cargo abundancewith
Conclusions
Explicit modeling of vesicular traffic reveals unappreciated
issues that question the validity of the widely accepted
Cdc42p-actin-vesicle positive feedback loop. Our models
suggest that to exploit positive feedback, a polarity regulator
would have to diffuse very slowly and be concentrated into
both exocytic and endocytic vesicles. At present, these
features are not known to apply to any yeast polarity regula-
tors. The modeling results should stimulate future studies to
determine whether polarity regulators with these characteris-
tics exist. If they do not, then our findings suggest that vesicle
traffic would act to dissipate, not reinforce, polarization.
Experimental Procedures
Simulating Bulk Traffic of Cdc42p
Simulations were performed with MATLAB using the program bulkcargo.m,
which models vesicle trafficking between an internal compartment and the
plasma membrane (all programs are provided in the Supplemental Informa-
tion; for parameter values, see Table 1).
The Plasma Membrane
We represented the plasma membrane of an unbudded diploid cell as
a square grid of 10,000 bins. Each bin contains Cdc42p at a concentration
that evolves with time as a result of diffusion, exocytosis, and endocytosis.
To avoid edge effects, we diffusionally connected the right and left edges of
the grid as well as the top and bottom edges.
The Intracellular Compartment
As in previous work, we considered the entire endomembrane system
relevant to Cdc42p recycling as a single well-mixed compartment. We set
the area of this compartment equal to that of the plasma membrane.
The Vesicles
Electron microscopy suggests a secretoryvesiclediameter ofw100nm[36]
and an endocytic vesicle diameter of w50 nm [37], with areas correspond-
ing to four bins and one bin of the plasma membrane, respectively.
Exocytosis
Exocyticeventsweremodeledasinstantaneoustransfersofmembrane and
Cdc42p from the internal compartment to the plasma membrane: a fusion
site on the plasma membrane was chosen (see below), and four (2 3 2)
bins were ‘‘inserted.’’ Because adding bins to the plasma membrane would
make diffusion difficult to model, we instead resized all bins to account for
the increased area. We redistributed the Cdc42p as follows: for the four ‘‘in-
serted’’bins, the Cdc42p concentration was reset tobeequal tothe Cdc42p
concentration in the internal compartment. The pre-exocytosis plasma
membrane was assumed to stretch radially, with the center of the four ‘‘in-
serted’’ bins as the origin. The Cdc42p in the (10,000 2 4) bins that corre-
spond to the pre-exocytosis plasma membrane was recomputed using
a spatial interpolation algorithm that preserves the preexisting Cdc42p
distribution around the insertion site and conserves the total amount of
Cdc42p (see Supplemental Results and Discussion for further discussion
of interpolation). The area and Cdc42p content of the internal compartment
were also adjusted to deduct the amount inserted into the plasma
membrane.
Endocytosis
Endocytic events were modeled as instantaneous transfers of membrane
and Cdc42p from the plasma membrane to the internal compartment:
a single bin on the plasma membrane was chosen to endocytose (see
below), and its membrane and Cdc42p content was removed from the
plasma membrane and added to the internal compartment. We resized
plasma membrane bins to account for the reduced total area and redistrib-
uted the Cdc42p by radial interpolation, maintaining the preexisting Cdc42p
distribution but deducting the endocytosed amount.
Vesicle Trafficking and Polarity
9
Please cite this article in press as: Layton et al., Modeling Vesicle Traffic Reveals Unexpected Consequences for Cdc42p-Mediated
Polarity Establishment, Current Biology (2011), doi:10.1016/j.cub.2011.01.012

Page 10

Frequency of Endocytosis
Endocytic events are marked by transient accumulation of F-actin in an
actin patch with a stereotypical lifetime of w15 s [31], and an unpolarized
cell of diameter w5 mm has w25 actin patches at any given time [38], sug-
gesting that endocytic events occur at a rate of w25/15 or 1.67/s. To a first
approximation, we assumed that they occur with similar frequency in polar-
ized cells.
Frequency of Exocytosis
Given that exocytic vesicles have four times the surface area of endocytic
vesicles, exocytosis must occur at 4-fold lower frequency to maintain
a steady state.
The Polarization Window
Wemodeledthewindow asanw1mmdiametercircularzoneinthecenterof
the plasma membrane [7].
Spatial Probability Distribution for Exocytosis
To simulate unpolarized cells, we assumed a uniform probability (1/10,000)
that any given bin is chosen as the exocytosis site. To simulate polarized
cells, we assumed that exocytosis occurs with uniform probability at any
bin within the window and does not occur elsewhere.
Spatial Probability Distribution for Endocytosis
To simulate unpolarized cells, we assumed a uniform probability that any
given bin is chosen as the endocytosis site. To simulate polarized cells,
we assumed that endocytosis occurs with a higher (uniform) probability
within the window and a lower probability elsewhere. Using m/n = 40 and
a 1 mm diameter window [7], w29% of all endocytic events occurred in
the window.
Diffusion
The diffusion constant for Cdc42p in the yeast plasma membrane is re-
ported to be 0.036 mm2/s [7]. In between exocytic and endocytic events,
we allowed diffusion to occur between neighboring bins in the plasma
membrane, using the backward Euler method to advance Cdc42p in
time.
Simulating Cargo Concentration into Exocytic Vesicles
Simulations were performed using the program exoconc.m, which is iden-
tical to the bulk cargo model except that exocytic events insert four bins
into the plasma membrane with 10 times the cargo concentration present
in the internal compartment, and the corresponding amount of cargo is sub-
tracted from the internal compartment.
Simulating Cargo Concentration into Endocytic Vesicles
Simulations were performed using the program fixedlifesink.m for the
uniform-time model and fillsink.m for the uniform-fill model. With the excep-
tions noted below, the models are similar to the exoconc.m model above.
Endocytic Patches as Diffusion Sinks
We assumed that cargo can diffuse into a bin designated as an endocy-
tosis site but cannot diffuse out. A problem in accurately modeling the
diffusion sink is that, because the cargo concentration in the sink may
be substantially higher than in the neighboring bins, the radial interpolation
algorithms we use to redistribute cargo following vesicle fusion and fission
events have the potential to artificially ‘‘move’’ cargo into or out of the sink.
To avoid this complication, we excluded the ‘‘sink’’ bins from the interpo-
lation process.
[35S]Pheromone Internalization
Simulations were initiated with uniform cargo concentration on the plasma
membrane and no cargo in the internal compartment. Endocytic events
transferred membrane but not cargo, to model the fact that labeled phero-
mone would be destroyed, not recycled.
v-SNARE Recycling in Unpolarized Cells
Exocytic vesicles were assumed to concentrate cargo 10-fold, and (for
t = 8 s, D = 0.0025 mm2/s) endocytic vesicles would concentrate cargo
w6-fold, so that at steady state, the internal compartment had a concentra-
tion 60% of that on the plasma membrane. To accommodate experimental
findings suggesting that w30% of the total v-SNARE is internal at steady
state [24], we set the size (area) of the internal compartment to be 70%
that of the plasma membrane. The unpolarized steady state was then
used as a starting point for simulations of polarized traffic.
Septin-Biased Endocytosis
We designated a septin ring zone as an annulus surrounding the central
window (inner diameter 1 mm, outer diameter 1.6 mm). Simulations specified
the probability that a given endocytic patch would (asa result of septin bias)
form inside the ring (Figure 4J). Septin-biased patches occur with uniform
probability at any bin within the ring, and other patches occur with uniform
probability anywhere.
Cdc42p as a Passenger Cargo
We simulated v-SNARE traffic with the uniform-fill model and 50% septin-
biased endocytosis but added a second cargo with D = 0.036 mm2/s to
represent Cdc42p. Each vesicle carries both Cdc42p and v-SNARE, but
only the v-SNARE influences the timing of internalization: if Cdc42p reaches
itsfilllevelfirst, thennofurtherCdc42ptrappingoccurs; ifv-SNARE reaches
its fill level first, then whatever amount of Cdc42p has accumulated at that
point is internalized.
Yeast Strains
The yeast strains used in this study were in the YEF473 background
(his3-D200 leu2-D1 lys2-801 trp1-D63 ura3-52). The rsr1::TRP1, ABP1-
mCherry:KanR[28], cdc42::HIS3, and GFP-CDC42:URA3 [39] alleles have
beendescribed previously. EDE1-GFP was generated using the PCR-based
C-terminal tagging method [40]. DLY12709 (MATa/a cdc42::HIS3/CDC42
GFP-CDC42:URA3/URA3 rsr1::TRP1/RSR1) was imaged for Figure 1, and
DLY12452 (MATa EDE1-GFP:KanRABP1-mCherry:KanRrsr1::TRP1) was
imaged for Figures 4G and 4I. For Figure 4H, DDY3868 (MATa EDE1-
GFP::HIS3 ABP1-RFP::HIS3) was used to determine Ede1p patch lifetimes
[17], and DDY3058 (MATa ABP1-RFP::HIS3) was used to calculate Abp1p
patch lifetimes [41].
Live-Cell Imaging and Image Analysis
Cells growing exponentially in 2% dextrose complete synthetic medium
were mounted on a slide with a slab of medium solidified with 2% agarose
(Denville Scientific). Images were acquired using an Axio Observer.Z1 (Carl
Zeiss) with a 1003/1.47 Plan Apochromat oil-immersion objective and the
stage incubator set at 30?C. Images were captured with either a QuantEM
EM-CCD camera (Figure 1) (Princeton Instruments) or a CoolSNAP ES2
CCD camera (Figure 4) (Photometrics) controlled by MetaMorph software
(Universal Imaging). All cells were imaged for fluorescence and differential
interference contrast with z planes at 0.24 mm step size.
Image deconvolution was performed with Huygens Essential software
(Scientific Volume Imaging) using the classic maximum-likelihood estima-
tion and predicted point-spread function. The line scans in Figure 1E were
generated using MetaMorph from the average intensity of a two-pixel-
wide line drawn around the periphery of an unbudded cell in the single
deconvolved z plane with the peak polarized intensity. The images in Fig-
ure 4 are maximum-intensity projections of 30 deconvolved z planes.
Images were prepared for presentation using Adobe Photoshop.
We used different microscopy methods to monitor Ede1p and Abp1p
patch lifetimes. Abp1p patches are short lived, so patch lifetime could be
assessed accurately by taking short (90 s) movies at high temporal resolu-
tion (1 frame/s) imaging a medial focal plane with an Olympus IX71 micro-
scope. Images were processed using ImageJ (http://rsbweb.nih.gov/ij/
index.html) before analysis. Because Ede1p patches are more abundant
and longer lived, it is difficult to resolve the crowded patches using this
approach, and longer movies are necessary to capture Ede1p lifetimes.
Thus, we imaged the top surface of the cells using near-total internal reflec-
tion fluorescence microscopy (near-TIRFM) to clearly distinguish individual
patches and took longer (4 min) movies at lower temporal resolution
(1 frame/2 s) to reduce photobleaching. Near-TIRFM is similar to TIRFM
[42], except that the angle of incidence is decreased in order to increase
the illumination depth beyond the evanescent wave [43, 44]. This method
better illuminates the plasma membrane through the yeast cell wall. Yeast
weregrowntolog phaseinsyntheticmediumlackingtryptophan andimmo-
bilized on concanavalin A-coated coverslips (Olympus). An Olympus IX81
microscope was used for simultaneous two-color TIRF microscopy of cells
expressing Ede1p-GFP and Abp1p-RFP, as described in [17]. Polarization
of the cells was assessed by the distribution of Abp1p-RFP patches visual-
ized in the red channel.
Supplemental Information
Supplemental Information includes Supplemental Results and Discussion,
five figures, and Supplemental Experimental Procedures and can be found
with this article online at doi:10.1016/j.cub.2011.01.012.
Acknowledgments
We thank Nick Buchler, Pat Brennwald, Tim Elston, Meng Jin, Mike Reed,
DanKiehart, andmembersofthe Lew laboratory forstimulatingdiscussions
and comments on the manuscript. We also thank Trevin Zyla for assistance
with strain construction. This work was supported by National Science
Current Biology Vol 21 No 3
10
Please cite this article in press as: Layton et al., Modeling Vesicle Traffic Reveals Unexpected Consequences for Cdc42p-Mediated
Polarity Establishment, Current Biology (2011), doi:10.1016/j.cub.2011.01.012

Page 11

Foundation grant DMS-0701412 to A.T.L. and National Institutes of Health
grants GM62300 to D.J.L. and GM50399 to D.G.D.
Received: October 25, 2010
Revised: December 30, 2010
Accepted: January 5, 2011
Published online: January 27, 2011
References
1. Pringle, J.R., Bi, E., Harkins, H.A., Zahner, J.E., De Virgilio, C., Chant, J.,
Corrado, K., and Fares, H. (1995). Establishment of cell polarity in yeast.
Cold Spring Harb. Symp. Quant. Biol. 60, 729–744.
2. Park,H.O.,andBi, E.(2007). Central roles ofsmall GTPasesinthe devel-
opment of cell polarity in yeast and beyond. Microbiol. Mol. Biol. Rev.
71, 48–96.
3. Irazoqui, J.E., Gladfelter, A.S., and Lew, D.J. (2003). Scaffold-mediated
symmetry breaking by Cdc42p. Nat. Cell Biol. 5, 1062–1070.
4. Kozubowski, L., Saito, K., Johnson, J.M., Howell, A.S., Zyla, T.R., and
Lew, D.J. (2008). Symmetry-breaking polarization driven by a Cdc42p
GEF-PAK complex. Curr. Biol. 18, 1719–1726.
5. Goryachev, A.B., and Pokhilko, A.V. (2008). Dynamics of Cdc42 network
embodies a Turing-type mechanism of yeast cell polarity. FEBS Lett.
582, 1437–1443.
6. Wedlich-Soldner, R., Altschuler, S., Wu, L., and Li, R. (2003).
Spontaneous cell polarization through actomyosin-based delivery of
the Cdc42 GTPase. Science 299, 1231–1235.
7. Marco, E., Wedlich-Soldner, R., Li, R., Altschuler, S.J., and Wu, L.F.
(2007). Endocytosis optimizes the dynamic localization of membrane
proteins that regulate cortical polarity. Cell 129, 411–422.
8. Wedlich-Soldner, R., Wai, S.C., Schmidt, T., and Li, R. (2004). Robust
cell polarity is a dynamic state established by coupling transport and
GTPase signaling. J. Cell Biol. 166, 889–900.
9. Slaughter, B.D., Das, A., Schwartz, J.W., Rubinstein, B., and Li, R.
(2009). Dual modes of cdc42 recycling fine-tune polarized morphogen-
esis. Dev. Cell 17, 823–835.
10. Adams, A.E.M., and Pringle, J.R. (1984). Relationship of actin and
tubulin distribution to bud growth in wild-type and morphogenetic-
mutant Saccharomyces cerevisiae. J. Cell Biol. 98, 934–945.
11. Kilmartin, J.V., and Adams, A.E.M. (1984). Structural rearrangements of
tubulin and actin during the cell cycle of the yeast Saccharomyces.
J. Cell Biol. 98, 922–933.
12. Valdez-Taubas, J.,andPelham,H.R.(2003).Slowdiffusionofproteins in
the yeast plasma membrane allows polarity to be maintained by endo-
cytic cycling. Curr. Biol. 13, 1636–1640.
13. Balch, W.E., McCaffery, J.M., Plutner, H., and Farquhar, M.G. (1994).
Vesicular stomatitis virus glycoprotein is sorted and concentrated
during export from the endoplasmic reticulum. Cell 76, 841–852.
14. Bonifacino, J.S., and Traub, L.M. (2003). Signals for sorting of trans-
membrane proteins to endosomes and lysosomes. Annu. Rev.
Biochem. 72, 395–447.
15. Sorkin, A. (2004). Cargo recognition during clathrin-mediated endocy-
tosis: A team effort. Curr. Opin. Cell Biol. 16, 392–399.
16. Maldonado-Ba ´ez, L., and Wendland, B. (2006). Endocytic adaptors:
Recruiters, coordinators and regulators. Trends Cell Biol. 16, 505–513.
17. Stimpson, H.E., Toret, C.P., Cheng, A.T., Pauly, B.S., and Drubin, D.G.
(2009). Early-arriving Syp1p and Ede1p function in endocytic site place-
ment and formation in budding yeast. Mol. Biol. Cell 20, 4640–4651.
18. Jenness, D.D., and Spatrick, P. (1986). Down regulation of the alpha-
factor pheromone receptor in S. cerevisiae. Cell 46, 345–353.
19. Rohrer, J., Be ´ne ´detti, H., Zanolari, B., and Riezman, H. (1993).
Identification of a novel sequence mediating regulated endocytosis of
the G protein-coupled alpha-pheromone receptor in yeast. Mol. Biol.
Cell 4, 511–521.
20. Schandel, K.A., and Jenness, D.D. (1994). Direct evidence for ligand-
induced internalization of the yeast alpha-factor pheromone receptor.
Mol. Cell. Biol. 14, 7245–7255.
21. Hicke, L., Zanolari, B., and Riezman, H. (1998). Cytoplasmic tail phos-
phorylation of the alpha-factor receptor is required for its ubiquitination
and internalization. J. Cell Biol. 141, 349–358.
22. Toshima, J.Y., Nakanishi, J., Mizuno, K., Toshima, J., and Drubin, D.G.
(2009). Requirements for recruitment of a G protein-coupled receptor
to clathrin-coated pits in budding yeast. Mol. Biol. Cell 20, 5039–5050.
23. Lewis, M.J., Nichols,B.J.,Prescianotto-Baschong,C., Riezman, H., and
Pelham, H.R. (2000). Specific retrieval of the exocytic SNARE Snc1p
from early yeast endosomes. Mol. Biol. Cell 11, 23–38.
24. Galan, J.M., Wiederkehr, A., Seol, J.H., Haguenauer-Tsapis, R.,
Deshaies, R.J., Riezman, H., and Peter, M. (2001). Skp1p and the
F-box protein Rcy1p form a non-SCF complex involved in recycling of
the SNARE Snc1p in yeast. Mol. Cell. Biol. 21, 3105–3117.
25. Qiu, W., Neo, S.P., Yu, X., and Cai, M. (2008). A novel septin-associated
protein, Syp1p, is required for normal cell cycle-dependent septin cyto-
skeleton dynamics in yeast. Genetics 180, 1445–1457.
26. Gladfelter, A.S., Pringle, J.R., and Lew, D.J. (2001). The septin cortex at
the yeast mother-bud neck. Curr. Opin. Microbiol. 4, 681–689.
27. Caudron, F., and Barral, Y. (2009). Septins and the lateral compartmen-
talization of eukaryotic membranes. Dev. Cell 16, 493–506.
28. Howell, A.S., Savage, N.S., Johnson, S.A., Bose, I., Wagner, A.W., Zyla,
T.R., Nijhout, H.F., Reed, M.C., Goryachev, A.B., and Lew, D.J. (2009).
Singularity in polarization: Rewiring yeast cells to make two buds. Cell
139, 731–743.
29. Ayscough, K.R., Stryker, J., Pokala, N., Sanders, M., Crews, P., and
Drubin, D.G. (1997). High rates of actin filament turnover in budding
yeast and roles for actin in establishment and maintenance of cell
polarity revealed using the actin inhibitor latrunculin-A. J. Cell Biol.
137, 399–416.
30. Kaksonen, M., Sun, Y., and Drubin, D.G. (2003). A pathway for associa-
tion of receptors, adaptors, and actin during endocytic internalization.
Cell 115, 475–487.
31. Kaksonen, M., Toret, C.P., and Drubin, D.G. (2006). Harnessing actin
dynamics for clathrin-mediated endocytosis. Nat. Rev. Mol. Cell Biol.
7, 404–414.
32. Boettner, D.R., D’Agostino, J.L., Torres, O.T., Daugherty-Clarke, K.,
Uygur, A., Reider, A., Wendland, B., Lemmon, S.K., and Goode, B.L.
(2009). The F-BAR protein Syp1 negatively regulates WASp-Arp2/3
complex activity during endocytic patch formation. Curr. Biol. 19,
1979–1987.
33. Rodal, A.A., Manning, A.L., Goode, B.L., and Drubin, D.G. (2003).
Negative regulation of yeast WASp by two SH3 domain-containing
proteins. Curr. Biol. 13, 1000–1008.
34. Ehrlich, M., Boll, W., Van Oijen, A., Hariharan, R., Chandran, K., Nibert,
M.L., and Kirchhausen, T. (2004). Endocytosis by random initiation
and stabilization of clathrin-coated pits. Cell 118, 591–605.
35. Loerke, D., Mettlen, M., Yarar, D., Jaqaman, K., Jaqaman, H., Danuser,
G., and Schmid, S.L. (2009). Cargo and dynamin regulate clathrin-
coated pit maturation. PLoS Biol. 7, e57.
36. Novick, P., Field, C., and Schekman, R. (1980). Identification of 23
complementation groups required for post-translational events in the
yeast secretory pathway. Cell 21, 205–215.
37. Prescianotto-Baschong, C., and Riezman, H. (1998). Morphology of the
yeast endocytic pathway. Mol. Biol. Cell 9, 173–189.
38. Karpova, T.S., McNally, J.G., Moltz, S.L., and Cooper, J.A. (1998).
Assembly andfunction of the
Relationshipsbetweencablesandpatches.J.CellBiol.142,1501–1517.
39. Bi,E.,Chiavetta, J.B., Chen,H.,Chen,G.C.,Chan,C.S.,andPringle,J.R.
(2000). Identification of novel, evolutionarily conserved Cdc42p-inter-
acting proteins and of redundant pathways linking Cdc24p and
Cdc42p to actin polarization in yeast. Mol. Biol. Cell 11, 773–793.
40. Longtine, M.S., McKenzie, A., 3rd, Demarini, D.J., Shah, N.G., Wach, A.,
Brachat, A., Philippsen, P., and Pringle, J.R. (1998). Additional modules
forversatileandeconomicalPCR-basedgenedeletionandmodification
in Saccharomyces cerevisiae. Yeast 14, 953–961.
41. Kaksonen, M.,Toret, C.P., andDrubin, D.G. (2005). Amodular design for
the clathrin- and actin-mediated endocytosis machinery. Cell 123,
305–320.
42. Newpher, T.M., Smith, R.P., Lemmon, V., and Lemmon, S.K. (2005). In
vivo dynamics of clathrin and its adaptor-dependent recruitment to
the actin-based endocytic machinery in yeast. Dev. Cell 9, 87–98.
43. Grigoriev, I., and Akhmanova, A. (2010). Microtubule dynamics at the
cell cortex probed by TIRF microscopy. Methods Cell Biol. 97, 91–109.
44. Reck-Peterson,S.L.,Derr,N.D.,andStuurman,N.(2010).Imagingsingle
molecules using total internal reflection fluorescence microscopy
(TIRFM). Cold Spring Harb Protoc 2010, pdb.top73.
45. Kim,H.B., Haarer,B.K., andPringle, J.R.(1991).Cellularmorphogenesis
in the Saccharomyces cerevisiae cell cycle: Localization of the CDC3
gene product and the timing of events at the budding site. J. Cell Biol.
112, 535–544.
actin cytoskeletonofyeast:
Vesicle Trafficking and Polarity
11
Please cite this article in press as: Layton et al., Modeling Vesicle Traffic Reveals Unexpected Consequences for Cdc42p-Mediated
Polarity Establishment, Current Biology (2011), doi:10.1016/j.cub.2011.01.012