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Function suggests nano-structure: towards
a unified theory for secretion rate,
a statistical mechanics approach
rsif.royalsocietypublishing.org
Ilan Hammel1 and Isaac Meilijson2
1
Sackler Faculty of Medicine, Department of Pathology, and 2Raymond and Beverly Sackler Faculty of Exact
Sciences, School of Mathematical Sciences, Department of Statistics and Operations Research, Tel Aviv University,
Tel Aviv 6997801, Israel
Research
Cite this article: Hammel I, Meilijson I. 2013
Function suggests nano-structure: towards a
unified theory for secretion rate, a statistical
mechanics approach. J R Soc Interface 10:
20130640.
http://dx.doi.org/10.1098/rsif.2013.0640
Received: 17 July 2013
Accepted: 9 August 2013
Subject Areas:
biophysics, biocomplexity
Keywords:
secretion rate, granule size, intracellular
calcium, SNARE, cellular communication
Authors for correspondence:
Ilan Hammel
e-mail: [email protected]
Isaac Meilijson
e-mail: [email protected]
Electronic supplementary material is available
at http://dx.doi.org/10.1098/rsif.2013.0640 or
via http://rsif.royalsocietypublishing.org.
The inventory of secretory granules along the plasma membrane can be
viewed as maintained in two restricted compartments. The release-ready
pool represents docked granules available for an initial stage of fast, immediate secretion, followed by a second stage of granule set-aside secretion pool,
with significantly slower rate. Transmission electron microscopy ultrastructural investigations correlated with electrophysiological techniques and
mathematical modelling have allowed the categorization of these secretory
vesicle compartments, in which vesicles can be in various states of secretory
competence. Using the above-mentioned approaches, the kinetics of single
vesicle exocytosis can be worked out. The ultra-fast kinetics, explored in
this study, represents the immediately available release-ready pool, in
which granules bound to the plasma membrane are exocytosed upon Ca2þ
influx at the SNARE rosette at the base of porosomes. Formalizing Dodge
and Rahamimoff findings on the effect of calcium concentration and incorporating the effect of SNARE transient rosette size, we postulate that secretion
rate (rate), the number (X ) of intracellular calcium ions available for fusion,
calcium capacity (0 M 5) and the fusion nano-machine size (as measured
by the SNARE rosette size K) satisfy the parsimonious M–K relation rate C [Ca2þ]min(X,M)e2K/2.
1. Introduction
Exocytic events of release-ready vesicle pool from secretory cells [1 –3] follow a
time course consistent with calcium concentration [4] and with Poisson behaviour [5,6], as evidenced by a number of statistical measures, such as the estimate
of its rate, the number of events recorded per unit time [6]. For vesicular
secretion, this direct measure exhibits a variety of statistical properties that
can be monitored. In particular, the probability of granule fusion with the
plasma membrane, which establishes the rate of secretion, is highly dependent
on random SNARE protein aggregation in a rosette at the porosome base [5],
arrival time and biochemical activity [1–6]. Based on statistical mechanics
modelling composed with meta-analysis of experimental data, we propose a
generalization of the common formula [4] for secretion rate, rate ¼ C [Ca2þ]M, expressed in terms of two key parameters that dictate secretion rate,
calcium concentration [Ca2þ] and the (maximal) calcium cooperativity (integer)
coefficient M (0 M 5), into the M–K relation: rate ¼ C1 E½ðC2 ½Ca2þ ÞminðX;MÞ C3K , where K is the (granule size-dependent) number of
SNAREs to form the secretion rosette, and X is the ( possibly random, calcium
concentration-dependent) actual cooperativity coefficient. Synaptic vesicle
release occurs spontaneously in the absence of Ca2þ binding at the calyx of
Held [7], in spite of low secretion rate and thus M can be 0. In previous
work, we have determined that K must be at least 1 [5].
Cellular communication depends on membrane fusion mechanisms. Fusion of a
secretory granule with the porosome at the cell plasma membrane [6–10] is
considered the critical step in regulated exocytosis. Granule fusion events, commonly quantified as a Poisson process [6,7], are controlled by the interplay
& 2013 The Author(s) Published by the Royal Society. All rights reserved.
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The authors have developed a growth and elimination (G&E)
model that describes a possible mechanism for basal granule
growth and secretion [5,16]. According to this model, homogeneous unit granules are packaged in the Golgi apparatus
(or otherwise), with some mean content (i.e. volume) y
and standard deviation s. Granules grow by fusing with
freshly packaged unit granules, or exit the cell. The rate
at which a granule of type Gn becomes of type Gnþ1 is ln ¼
lnb and the rate at which the granule exits the cell is mn ¼
mng. As usual with Markov modelling, these two competing
scenarios manifest themselves at independent exponentially
distributed times, and the earliest to happen, wins. The authors
have argued [5,16] that g ¼ 2 (2/3)(Kg 2 1) and b ¼ 2(2/
3)(Kb 2 1), where Kg is the number of SNARE protein complexes that constitute the rosette that docks a granule with
the cell membrane for exocytosis, and Kb is a physical or conceptual counterpart for homotypic fusion of a mature granule
and a unit granule. This model explains the quantal nature of
granule content, both in the cytoplasm and upon one-granule-at-a-time secretion. Quantal means [14–16] that granules
of type Gn have mean content ny and standard deviation of
pffiffiffi
content ns. Thus, quantal granule content distribution is multimodal, with equally spaced peaks. The measured
standardffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
deviation of the nth peak should behave such as ns2 þ t2 ,
where t is the error of measurement standard deviation [14–
16]. The packaging coefficient of variation s/y is of the order
of 10%, and the error of measurement coefficient of variation
is of the order of 10–30% [14–16]. A distinctive feature of the
G&E model is that it predicts the emergence of two distinct
granule volume distributions, the stationary distribution that
describes steady-state statistics in the cytoplasm and the exit
distribution that describes granule content distribution at
secretion [5,16]. These two distributions carve up quantal
nature, share the parameters y , s and t but differ in the mixture
weights at the various peaks. This statistical mechanics
approach views SNARE components as interacting particles,
and provides a simple mathematical explanation for a steadystate ‘fusion nano-machine’ of SNARE self-aggregation, that
predicts secretion rates as described above, proportional to a
term of the form C32K, as in the M–K rate relation above.
Based on simple Euclidean geometry, the authors have
argued [5] that the SNARE rosette size K should be proportional to the square root of granule diameter, and
estimated it empirically from
pffiffiffiffi the mean granule diameter (D,
nm) as Kg ¼ ð0:9 + 0:2Þ D. Statistical properties of evoked
exocytic events were checked over a wide range of granule
endocrine cells
e8
maximal rate of secretion (vesicles per second)
2.1. Dependence on the SNARE rosette petals
neurons
secretory cells
e6
e4
J R Soc Interface 10: 20130640
2. The model
2
e10
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between cellular Ca2þ signals and catalysed by specific proteins
(mainly SNAREs) [1–11]. The delivery of newly formed
secretory content to the cytoplasm granule inventory occurs
through a direct fusion of recently formed granules and mature
granules. The authors have documented [12–14] evidence that
newly formed granules are monomers of quantal size G1 and
larger granules are polymer (n-mer) progressive aggregates of
the granule monomer (Gn ¼ nG1). SNARE proteins play an
essential role in all intracellular fusion reactions associated with
the life cycle of secretory vesicles, whether vesicle–vesicle
(homotypic fusion, as just described) or vesicle–plasma membrane fusion (heterotypic fusion, leading to secretion).
e2
1
e–2
e–4
e–6
0
5
10 15 20 25 30 35 40
granule mean diameter (nm)
pffiffiffi
Figure 1. Scattergram analysis of granule size ( D, equivalent to SNARE rosette
size) as associated with maximal secretion rate. For a tabulated data list, see the
electronic supplementary material, table S1. The regression of ln(rate) on K 2 1
is 10.4893 2 0.4848 (K 2 1), or rate 36000 0.6158K21. The role of
Ca2þ should, in principle, explain the heterogeneity within the blocks shown in
the figure, but the crucial effect of K in the figure is well explained by the G&E
model described previously by the authors [5,16].
sizes (D ¼ 25–1270 nm, y ¼ 8 103 – 1.1 108 nm3), using
secretion rates and granule morphometric characteristics
recorded from published data (see electronic supplementary
material, table S1). Scattergram analysis of the natural logarithm of secretion burst rate versus estimated rosette size
confirms a linear relation (figure 1), leading very close to
rate 4.5 104 e2(K21)/2. Thus, the shape C2K
of basal
3
secretion rate is an accurate description under evoked
secretion as well, with C3 e0.5 ¼ 1.65. Interestingly, the
rosette size required for fusion has been claimed to be at least
three SNAREs [17], which would lead via the regression
above to a theoretical maximal achievable secretion rate of
13 600 vesicles s21, twice the maximal observed rate of about
6000 vesicles s21, that corresponds in the regression equation
to a rosette composed of five SNAREs. Perhaps, the minimal
in vivo rosette size is 4.
The literature of the past 40 years reports on many studies
in which secreted granule content has been carefully monitored, under two conditions: basal and evoked secretion. The
basal case should conform to the exit distribution described
above, and indeed, the authors have documented good correlation with the above set of equations [5,16]. The evoked case is
generally assumed to be the result of sharply increased Ca2þ
concentration, inducing the exocytosis of most or all granules
in the vicinity of the plasma membrane porosome complex
[6–13]. Porosomes are permanent cup-shaped lipoprotein
structures at the cell plasma membrane [8–10], with
t-SNAREs at its base. Secretory vesicles transiently dock and
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It has not eluded note that calcium ion is also a significant
parameter; after all, it has been documented for more than
3
J R Soc Interface 10: 20130640
2.2. The M –K rate event
half a century [4] that secretion rate is proportional to
[Ca]M, where [Ca] is Ca2þ concentration and M is the
number of cooperating ions (0 M 5), reflecting that multiple Ca2þ ions binding to M-sites are required for fusion
acceleration [2]. Such relation, well documented for exocytosis and recently confirmed also for homotypic fusion [25], is
not surprising; Ca2þ ions are the key ions required for the finalization of membrane fusion. In the case of Ca2þ ion binding,
the first Ca2þ ion has three to five different locations (depending on the ligand protein) where it can bind [2,8–13]. In case
M . 1, the reaction is positively cooperative, namely the first
Ca2þ ion binding to a site increases the protein–Ca2þ ion complex affinity at other binding sites. This represents a state of
higher entropy compared with binding the last Ca2þ ion,
which has only one location where it can bind. Thus, in
going from the unbound to the bound state, the first Ca2þ
ion must rise above a larger entropy change than the last
Ca2þ ion. In an attempt to represent a broader range of
secretion rates, the number X of cooperating Ca2þ ions
should be considered random, with M as an upper bound. It
is natural to think of X, the number of ions located in a small
vicinity of the binding protein, as being Poisson-distributed
[8,9] with some mean h. From this point of view, secretion
rate should be considered to follow the M–K rate relation,
whose M-dependence component is rate ¼ C1Eh[(C2
[Ca])min(X,M)]. The parameter C2, which adjusts for the units
in which [Ca] is measured, can be embedded in C1 only
when the exponent M is constant. Assuming a rosette diameter
of 6 nm and an effective distance between the two membranes
[9] of 0.5 nm (5 Å), the volume of the emerging cylinder
is 1.4 10220 cm3. Thus, one Ca2þ ion, that will be caged
within such narrow volume, will create a local concentration
of 8 mM which is about four magnitudes of the cytoplasmic Ca2þ ion concentration (30 nM) and close to normal
blood-ionized calcium level (1.9–2.7 mM). These significant
concentration gradients facilitate a very rapid increase in cytosolic calcium concentration upon opening of calcium channels.
Accordingly, there are two options for cytoplasmic Ca2þ ions
to diffuse into the effective fusion zone, either individually or
bound to a calcium-binding protein (e.g. synaptotagmin).
Both options have similar stochastic nature [11].
X-ray diffraction measurements demonstrated that v- and
t-SNAREs in opposing membranes overcome repulsive
forces, bringing them closer, so that the inter-bilayer space
between them is just 2.8 Å [22]. It was hypothesized that
hydrated calcium ions (having a shell of six water molecules
and measuring nearly 7 Å) could bridge the oxygen (and coordinated water molecules) of the opposing phospholipid head
groups, resulting in loss of water from the site, leading to
lipid mixing and fusion. It has been demonstrated that if
t-SNARE and v-SNARE vesicles are mixed prior to the
addition of calcium, the 2.8 Å gap between opposing membranes established by SNAREs would hinder the hydrated
calcium measuring nearly 7 Å to fit between the membranes
and bridge the opposing phospholipid head groups, preventing membrane fusion [23,24]. This, indeed, turned out to be
correct. The findings have further been confirmed by blind
molecular dynamics simulations using calcium, dimethylphosphate (smallest lipid) and water molecules, demonstrating the
bridging of the opposing oxygen atoms of dimethylphosphate
and loss of water from the bridged site. The distance between
the opposed oxygen atoms bridged by calcium is 2.92 Å, close
to the 2.8 Å determined from X-ray measurements. Therefore,
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fuse at the porosome base via a fusion pore formed by the
assembly of t-/v-SNARE complexes in a rosette pattern
[8–10]. In various secretory cells, the fusion pore may flicker
between closed and open states unless it fully expands
(permanent, complete fusion) or closes again (transient, kissand-run fusion) [8–13]. If this is the case, evoked secreted
granule content should follow the stationary distribution.
Strong early evidence in this direction was provided by
von Schwarzenfeld [18]. Von Schwarzenfeld observed and
measured that under increased neurosecretory cell activation,
the leading vesicles along the plasma membrane are exocytosed first. Similar morphological work on classical secretory
cells documented, in parallel to von Schwarzenfeld, that granules along the plasma membrane are secreted first, followed by
fusion of adjacent inner granules, forming a degranulation sac.
This fusion-related membrane advancement towards neighbouring granules by progressive burrows has been termed
compound exocytosis [19]. These steps correlate mainly with
Ca2þ flow into the cell upon activation.
Basal and evoked secretion states are exocytosis modes of
the same pool of granules located in close proximity to the
plasma membrane. Secreted granule content should display
a distribution that is a mixture of the two, ranging from the
exit distribution at the low Ca2þ concentrations typical of
basal secretion, to the stationary distribution, at the opposite
end of high Ca2þ concentration. The exit distribution EXIT
depends [5,16] on the two parameters g – b and m/l, and
the stationary distribution is given by STAT(n) ¼ C EXIT(n)n2g. As g , 0, the stationary distribution dominates
the exit distribution stochastically—smaller granules exit
basally and bigger granules exit evokedly [5,16].
It is conceivable that secretion rates in the two modes are
tightly coupled, determined principally by the size-dependent, mode-independent ability to form the SNARE
complex essential for secretion. The difference between the
two modes could be that in the basal mode a successful
SNARE rosette releases a single granule (and as such, behaving according to the EXIT distribution), whereas in the
evoked mode, once the rosette is effective, a number of granules (with typical stationary behaviour) are released via the
progressive burrows of compound exocytosis, all ‘hitchhiking’ on the original rosette. Thus, we expect evoked
secretion rates to decrease exponentially in the rosette size
K, as modelled by the M – K rate relation. The regression
line in figure 1 estimates the decay coefficient C3 to be 1.65.
The authors have paradigmatically identified [5] the two
secretion content distributions with the exit and stationary
distributions, and applied maximum-likelihood estimation
to 12 examples of published data, to estimate in each case
the parameter triple ( y , s and t) that applies simultaneously
to the two empirical distributions, as well as the quantal skeleton (mixture weights at the various peaks) corresponding to
each of the two [5,14,16]. The observation that b and g have
values in the range 22 to 26 led us to abandon attempts at
modelling growth and secretion rates by classical physics
such as van der Waals forces, in favour of particle physics
and statistical mechanics [5], taking SNARE proteins and
porosomes to centre stage [20– 24].
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As indicated above, in some cells, there is more than one type
of secretory granule. The different classes of granules are
mobilized in a hierarchy, seemingly adjusted to the different
roles these granule types play. Because, in all monitored cases,
the granules appear indiscriminately dispersed throughout the
cytosol in the cell, this hierarchy must rely on mechanisms that
discriminate between the different granule subsets. We suggest
that rosette size aggregation probability may dictate the hierarchy in tandem with calcium requirements (M–K rate relation).
Elevation of intracellular Ca2þ is known to evoke exocytosis of
storage granules, but the (hitherto unidentified) complete molecular machinery by which this occurs is the subject of
abundant research. We cannot foresee a simpler mechanism in
which calcium may invoke secretion hierarchy in the absence
of cooperation with SNARE rosette size. Indeed, Neher &
Sakaba [2] documented such a relation. These authors found
that when calcium ion concentration is elevated directly at
the calyx of Held presynaptic terminal with the use of caged
calcium, cleavage of SNAP-25 by botulinum toxin A produces
a strong reduction in the calcium sensitivity for release [26].
Thus, fast and slow components of neurotransmitter release
can be distinguished (see electronic supplementary material,
figure S3). The M–K rate relation can shed insight on rosette
3.1. Granule size affects exit rate
The M–K statistical mechanics-based equation views the
SNARE units as an integer number of undifferentiated particles. Secretion diversity mechanisms in a variety of cell
types depend on the molecular subfamily heterogeneity of
SNARE proteins. Different combinations of SNARE complexes
may constitute the rosette for a given vesicle. The diversity of
(K) SNARE regulatory proteins and of (min(X,M)) Ca2þ
sensor sites on various proteins can induce and regulate
fusion. Such diversity may explain the rate variability observed
in the endocrine cells, with granule diameter range
, 250 nm). The rate ratio for such range (e5 150)
(150 , D
is equal to a calcium concentration factor increase of 5–12
(M ¼ 3 or 2, respectively). This model describes the critical
impact of SNARE rosette size and local Ca2þ concentration
on secretion rate. The M–K relation suggests that vesicle priming is the key step for secretion. It has been documented that
primed granules fuse immediately as a response to increase
in cytoplasmic Ca2þ level (mainly along the plasma membrane). This fusion event is mediated directly by the SNAREs
and driven by the energy provided from SNARE assembly.
The calcium-sensing trigger for this event is the calciumbinding protein (e.g. calcium sensor synaptotagmin). Granule
docking is pre-stage to priming. The SNARE proteins do not
appear to be involved in the docking step of the granule lifetime cycle [12,13]. Sensitivity towards intracellular Ca2þ as a
signal to elicit organization and mobilization has been
observed to vary among the different granule subsets, in
agreement with the hierarchy of their mobilization [28]. Polymorphonuclear cells rapidly secrete their content, inducing
non-specific tissue destruction. As has been well recognized,
major distinctions happen between the three different neutrophil granule subsets regarding the extent to which these are
mobilized both in vitro and in vivo: gelatinase positive granules
(major axis ¼ 209 nm; minor axis ¼ 85 nm) are secreted more
readily than specific lactoferrin positive granules (major
axis ¼ 305 nm; minor axis ¼ 124 nm), which again are exocytosed more readily than azurophil myeloperoxidase positive
. 400 nm) [28–30]. Proteases from azurophil
granules (D
granules may activate the cathelicidin present in the specific
granules by proteolytically removing the inhibitory (and protective) N-terminal cathelin-like part of the protein. Similarly,
gelatinase and collagenases may be converted from their zymogen to active proteases by elastase liberated from azurophil
granules [28–30]. As such, the small granules containing the
substrates are secreted first, whereas the activating enzymes
(stored within the large granule) are secreted last. As indicated
in previous works [5,16], smaller granules have higher
4
J R Soc Interface 10: 20130640
3. Discussion
size, estimating it to be about six to eight SNARE units. Because
the mean secretory vesicle diameter is 46 nm, the estimated
number of SNAREs is 6 (K ¼ 0.9 460.5 6.1), in good agreement with experimental data. Fluorescent proteins can be
genetically fused to target proteins (e.g. SNARE proteins)
and consequently offer direct access to specific and stoichiometric aggregation. Stochastic optical reconstruction
microscopy analysis combined with quantitative clustering
algorithms showed clustering of syntaxin and SNAP-25 molecules [27]. In addition, the data show that small clusters can
merge with other clusters, to form super-clusters (above 60
SNAREs). Our model suggests that the small clusters might
serve as the basic units to form the rosette.
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the participation of calcium-binding proteins such as synaptotagmin provides in cells further regulation to the membrane
fusion process. This is consistent with the role of the higher
local membrane curvature and higher accumulated energy
during fusion pore expansion for smaller vesicles.
We propose now to relate the global SNARE protein particle
dynamics that differentiate between cell types with the local
Ca2þ ion cooperation dynamics, to present a unified theory of
secretion rate, under which secretion rate is representable as
the product of two contributions: the local ion cooperation
effect C1(C2[Ca])min(X;M) and the global SNARE effect C2K
3 .
The unified representation for secretion rate is then the M–K
rate relation rate C1Eh[(C2 [Ca])min(X,M)]C2K
3 .
Even in the absence of secretory cell activation (e.g. action
potentials), most cells basally secrete individual granules
[1–3,6]. The contribution of Ca2þ ion for such events has
been less investigated in vivo, although, as has been documented [20–26], Ca2þ ion was observed in liposome experiments
to be mandatory for the fusion process, but not evidently in
a cooperative manner (namely M ¼ 1 is not excluded)
[7,20,21,25]. This issue can be investigated by means of statistical analysis based on the G&E theoretical model, via the
estimation of the effective kinetics factor (m/l ), the relative rate
of granule membrane fusion to granule homotypic fusion.
Electronic supplementary material, figures S1 and S2, supports
the Ca2þ ion-dependent mechanism. It is hard to think of any
other ligand-associated molecule that can modify the fusion
process along two to three orders of concentration. Taken
together, secretion rate, calcium concentration and rosette
size are postulated to satisfy rate ¼ C1E[(C2[Ca])min(X,M)]C2K
3
(with C3 e0.5 ¼ 1.65), the parsimonious relation that holds
through all activation states. Figure 1 (and electronic supplementary material, figure S2) exhibits separate evidence for
the M and K effects, and we are unaware of data in support
of their joint effect. The cell should adapt M and K as dictated
by the activation process, a subject for future research.
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speed of cAMP action during high-glucose stimulation, and
found that the augmentation of exocytosis by cAMP occurred
within a fraction of a second for SVs but with a delay of 5 s
for LVs. Pheochromocytoma PC12 cells provide a definitive
example of such a mechanism. In these cells, acetylcholinecontaining small clear vesicles (SVs) and monoaminecontaining large dense-core vesicles (LVs) undergo exocytosis
with time constants of about 30 ms and 10 s, respectively
[30 –33]. In spite of the specificity of their cargoes and of
their intracellular pathways, these secretory vesicles share
similar membrane components, and their exocytosis is
mediated by the same SNARE system.
rsif.royalsocietypublishing.org
probability of basal release. To decrease basal secretion of
harmful proteins, the neutrophils terminate granule synthesis
(increase l and thus decrease m/l ) and generate oblate granules
(and thus increase K).
Hatakeyama et al. [31] investigated exocytosis in pancreatic b-cells with two-photon extracellular polar-tracer imaging.
These authors detected marked Ca2þ-dependent exocytosis
of small vesicles (SVs) with a mean diameter of 80 nm in
addition to exocytosis of some large vesicles (LVs). Exocytosis
of SVs occurred with a time constant of 0.3 s, whereas that of
LVs showed a time constant of more than 1 s. Furthermore,
they applied photolysis of caged cAMP to quantify the