Abstract
One difficulty in conducting biologically meaningful dynamic analysis at the systems biology level is that in vivo system regulation is complex. Meanwhile, many kinetic rates are unknown, making global system analysis intractable in practice. In this article, we demonstrate a computational pipeline to help solve this problem, using the exocytotic process as an example. Exocytosis is an essential process in all eukaryotic cells that allows communication in cells through vesicles that contain a wide range of intracellular molecules. During this process a set of proteins called SNAREs acts as an engine in this vesicle-membrane fusion, by forming four-helical bundle complex between (membrane) target-specific and vesicle-specific SNAREs. As expected, the regulatory network for exocytosis is very complex. Based on the current understanding of the protein-protein interaction network related to exocytosis, we mathematically formulated the whole system, by the ordinary differential equations (ODE). We then applied a mathematical approach (called inverse problem) to estimating the kinetic parameters in the fundamental subsystem (without regulation) from limited in vitro experimental data, which fit well with the reports by the conventional assay. These estimates allowed us to conduct an efficient stability analysis under a specified parameter space for the exocytotic process with or without regulation. Finally, we discuss the potential of this approach to explain experimental observations and to make testable hypotheses for further experimentation.
Introduction
Exocytosis is the fundamental physiological process that leads the traffic of vesicles to fuse with the plasma membrane, releasing its vesicle contents into targeted cells that control many cellular processes [1]–[3]. Substantial studies have shown that it involves multiple steps from vesicle trafficking, docking, priming to fusion [1]–[14]. During this process, a set of proteins called SNARE proteins occupy a central position in the fusion by protein-protein interacting between vesicular-specific and (membrane) target-specific SNARE protein isoforms, denoted by vSNARE and tSNARE, respectively. Moreover, this SNARE-mediated fusion is highly regulated through different modes [6], [15]–[23]. For instance, one mode is through the protein-protein interaction with MUNC18, a member of Sec1/Munc18 (SM) protein family, while the other mode is through the Ca
-triggered exocytosis [14], [15], [5], [20].
Although experimental studies have provided invaluable insights for the underlying exocytosis mechanisms, the process of exocytosis is a typical example to show the difficulty in conducting an analysis at the systems biology level [24]–[28]. That is, while the biochemical reaction chain is straightforward and simple, the regulation in vivo of the system is complex. As many kinetic rates are unknown, and concentrations of proteins, complexes and substrates keep changing in both vivo and vitro environments, a biologically meaningful, global system analysis is intractable in practice. Earlier, Mezer et al. [10] proposed a computational platform to model the exocytotic process. They formulated these protein interactions into a sequential (feed-forward only without any regulation) interaction pathway to describe the exocytotic system dynamics.
In this paper, we utilize the exocytotic process as a model system to present a computational framework for system modeling and analysis. Similar to [10], we model the dynamics and architecture of the complex system by the ordinary differential equations (ODEs). First, we model the whole system by taking the regulatory elements into account. Second, we use a math techniques called inverse problem to estimate the rate parameters for the basic steps of biochemical reactions. Through the method, we are able to recover and optimize these parameters based on limited in vitro experimental data. Third, based on the above estimates, we can therefore approximately study the stability behavior of this system with and without MUNC18 regulation. We then attempt to explain experimental observations about different fusion efficiency caused by the change of SNARE proteins’ concentration and multiple complexes in the SNARE-induced membrane fusion. Moreover, we make a few interesting predictions that can be verified by further experimentations.
Results and Discussion
The Protein Interaction Network of Exocytosis
From the view of gene network, the exocytotic process is a sophisticated combination of sequential interactions of well-defined proteins and protein complexes [1]–[13]. As shown in Fig.1, it has three major components. The first step of the basic reaction component includes two membrane proteins, SNAP25 (synaptosome-associated protein, 25 kDa) and syntaxin, together forming the so-called tSNARE; here 
means target, the plasma membrane where the vesicle is heading for. Another important protein is vesical-associated membrane protein (VAMP2), belonging to the category of vSNARE (vesicle). In the second step, the protein complex formed by tSNARE and vSNARE is the fundamental step for the membrane fusion. In our study we consider two regulatory components, which are MUNC18-mediated and Ca
-dependent regulation pathways, respectively. On the other hand, from the view of systems biology the mechanism of this exocytotic process is a dynamics system capturing the temporal change of the concentrations of proteins and intermediate complexes, which can be formulated based on an ODE dynamic system, as shown below in details.
Figure 1. The whole process of fusion used in the mathematical model is shown.
One direction arrows and symbol of 
represent the reaction between proteins, ions and complexes, while full direction arrows connect two parts of a single reaction. Modified from [1]–[6].
The basic steps
The well-known foundations [4], [3], [8] for this exocytotic processes are the following two reactions.
 |
 |
where the protein complex FHC stands for the four-helical bundle. Formation of FHC complex is the main step to promote membrane fusion, an essential part of exocytosis. In addition, there are several follow-up complex modifications. For instance, the function of complexin is as a clamp [20], resulting in
 |
where FC is the generic notation for the protein complex of FHC and complexin.
-dependent regulation.
is the main trigger for the initiation of intracellular exocytosis [4], [14], [15], [5]. Suggested by [16], [4], the regulation of 
is executed through stimulating synaptotagmin. A well-known mechanism is that 
binds with the SNARE complexes (FHC) and stimulates the fusion [2]. The reaction equations to characterize the mechanism regarding 
and synaptotagmin are given by
 |
where CaS stands for the complex of synaptotagmin and one Ca
ion, and
 |
 |
 |
where the generic notation Tsc represents the protein complex of tSNARE and synaptotagmin binding four Ca
ions, and FHC
represents the complex of FHC and synaptotagmin binding four Ca
ions.
MUNC18-dependent regulation
MUNC18 is an important regulatory protein for the exocytotic system [6], [17], [22], through two different modes: (
) MUCNC18 associates with syntaxin to remove them from the assembly into the SNARE complexe at the beginning stage; and (
) MUNC18 stimulates the fusion process by associating with FHC. These two reaction mechanisms can be written as follows.
 |
where FHC
is the complex of MUNC18 and FHC to help the fusion process; and
 |
where Smc is the generic name for the protein complex of MUNC18 and syntaxin.
Mathematical Modeling for the Whole Exocytotic System
Putting together, we have formulated a mathematical model by the ordinary differential equations (ODE) to capture how the concentrations of different proteins and complexes vary with time and how they interact each other. Based on the law of mass action and Michaelis-Menten Kinetics, and using the conventional notation 
for the concentration, the ODE system is given by.
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
(1) |
One may raise the question, due to the complexity of this network, whether we have empirical evidence enough to show the concept we try to put forward. In a recent article, we [29] have conducted a comparative network motif analysis for the Sec1/Munc18-SNARE regulatory mechanisms through a comprehensive compile of experimental data from different species and different cell types. In spite of some differences in details that have been shown important for cell-specific and species-specific system behaviors, we confidently conclude that Eq.(1) may conceptually represent the basic dynamic system that is likely universal. Some comments about Eq.(1) are presented below.
Dynamics of 
The dynamics for 
in exocytosis is complex. To be analytically feasible, we assume that during the fusion process, concentration of 
ions at active zone is temporal dependent. Thus, the dynamics of 
ions around the region of fusion (active zone) can be characterized as.
 |
 |
 |
 |
(2) |
where 
is the recruitment source of calcium. Nevertheless, the in vivo concentration of Ca
ions may stay at a roughly constant level as both external and internal sources may have kept the balance of Ca
. In this case, Eq.(2) can be replaced by the simplest form 
Constant.
Self-association of syntaxin
We notice that self-association of syntaxin is possible such that 
, where 
, syntaxin
represents the complexes made of 
syntaxins [12]. Hence, if we take the effect of self-association into account, the system of Eq.(1) needs to be modified as follows: we have the equation for the concentration of syntaxin.
 |
 |
 |
(3) |
and additional four equations to describe the dynamics of self-associated complexes, that is,
 |
(4) |
where 
Mass conservation
One can show that the ODE system of Eq.(1) complies with the detailed balance principle and the mass conservation. For instance, because the only products of the reactions involving MUNC18 are Smc and FHC
, the change of concentration of MUNC18 is only relevant to the concentrations of these two complexes. Indeed, for the subsystem that only involves MUNC18, Smc and FHC
, we obtain.
 |
 |
(5) |
since there are no Smc and FHC
initially.
Spatial effect
Denote all of the variables (concentrations) in Eqs.(1)–(5) by a vector 
so that the ODE system can be rewritten in a concise form of 
, where 
is vector of functions on the right hand side of each equation. If the spatial effects of proteins and protein complexes are considered, this system should be generally written as follows.
 |
(6) |
where 
stands for the vector of diffusion coefficients of proteins, complexes and ions. Study of reaction diffusion equations described by Eq.(6) would be interesting particularly for the problems related to the developmental process.
Estimation of Reaction Rate Parameters
The whole system for the exocytotic process as described in Eq.(1) is a typical example to show the general difficulty in systems biology [24]–[28]. While the biochemical reaction chain is simple, the regulation in vivo of the system can be very complex. In addition, most paremeters remain unknown in this ODE system, including the initial concentrations of different proteins and complexes, and the reaction rates in both vivo and vitro environments. Hence, it is almost intractable in practice to carry out a global system analysis. As a first step to overcome this difficulty, we attempt to estimate the rate parameters for the basic steps of exocytotic process. Among different methods, we choose the technique of inverse problem that has two advantages: the required data size is small, and the algorithm guarantees the uniqueness and efficiency [25].
The fundamental subsystems
As the well-known machinery, chemical reactions.
 |
are fundamental for membrane fusion. The behavior of this subsystem involving only proteins SNAP25, syntaxin and VAMP2 can be described as
 |
 |
 |
 |
 |
 |
(7) |
Using the inverse problem technique [30] to estimate rate parameters requires initial concentrations. In the following the symbol 
is used for the concentration of variable 
at time 
. From the experimental data [9], we set the initial condition for system Eq.(10) to be: 
, and 
. It should be noticed that the estimation of kinetic rate parameters are usually insensitive to the initial conditions, as verified by our simulation studies (not shown).
Reparameterization for data-fitting
In the experimentation, researches use the fluorescence intensity, 
to measure the time-dependent fusion process. The relationship between the concentrations of core complexes (FHC, FHC
, FHC
) and the fluorescence intensity, 
, needs to be addressed in some details. Some experimental studies such as [11] suggested that the function 
can be roughly considered to be linear when the signal strength is far below the saturated level. In this case we have.
 |
(8) |
where 
are unknown constants. We further assume that the generated intensity of fluorescence due to fusion is 
and the measured intensity of fluorescence is 
, where 
is a constant supply for fluorescence, resulting in
 |
(9) |
Employing Eqs.(8)-(9) in the system Eq.(7), denote 
, 
, 
, 
, 
(or 
), we have
 |
 |
 |
 |
 |
(10) |
where 
and 
. Thus, parameter recovery for the fundamental subsystem equivalent to identify the parameters 
of Eq.(10).
Estimation by the inverse problem algorithm
To recover the appropriate reaction rates, we apply technique introduced by [27] to Eq.(10). Some useful theorems are presented in the section of Materials and Methods. Using the data from [9], the identified parameters are shown in the table 1. We compare the numerical results based on the identified parameters with experimental data in Fig.2, and the error is 
.
Table 1. Reaction rates for the fundamental subsystem.
Note: 
is the reaction rate for 
; 
is for 
; 
is for 
, 
is for 
, and 
is the fusion-concentration constant.
Figure 2. A comparison to the good-of-fit level between the numerical results by the inverse problem analysis and the original experimental data from [9].
The error is about 
.
Stability Analysis of the Fundamental Subsystem
Estimation of rate parameters of the subsystem Eq.(10), as summarized in Table 1, allows us to carry out the stabilizing analysis under a specified parameter space. Considering the subsystem Eq.(10) with 
instead of 
, we first study the fundamental subsystem without any regulation, under the initial concentrations 
, 
, 
, 
and 
for proteins SNAP25, Syntaxin, and VAMP2, and protein complexes tSNARE and FHC, respectively. While the formal mathematical treatment is shown in the section of Data and Methods, below we discuss about the biological interpretations.
Our analysis has shown that the final steady state level of the fusion is highly dependent on initial concentrations. Obviously, three proteins (SNAP25, syntaxin and VAMP2) must exist at 
so that 
, 
and 
. It is reasonable to assume no any fusion (here measured by FHC) at the initial time point, which means 
. The only case we have to deal with carefully is the initial concentration of tSNARE complex, 
. This is because in vivo, tSNARE is already preformed in the plasmic membrane; and then carried by vesicles, vSNARE (VAMP2 in our case) binds with it to generate fusion. In this sense, we assume 
in general.
To be concise, we define 
, 
, and 
and 
. Let 
(
) and 
(
) be the steady-states for 
, 
, 
, 
and 
, respectively, and the steadt state vector 
. Our goal is to obtain the analytical form of 
. As shown in the section of Data and Methods, our mathematical analysis considers three cases under the specified parameter space given by Table 1.
Case-A assumes that the initial concentration of SNAP25 and syntaxin are the same such that
. Denote 
, provided 
, we have shown there are two locally stable-steady states, denoted by 
and 
, respectively, corresponding to 
or 
. If 
, the degenerated steady state 
is also stable.Case-B studies the problem without the assumption of same initial concentration of SNAP25 and syntaxin. Our stability analysis shows that, provided
where 
, there are four steady states of 
that are locally stable, corresponding to (
) 
and 
, (
) 
and 
, (
) 
and 
, and (
) if 
, 
, and 
, respectively.Case-C considers a more general case that the reaction ratio
and concentrations of SNARE proteins and complexes are in the same order, that is, 
and 
. It has been shown taht the steady-states are locally stables under the following conditions: (
) 
and 
; (
) 
, 
and 
; and (
) 
and 
, respectively.
Since we are mostly interested in the final steady state-level of fusion, i.e., 
, the biological meaning of above stability analyses can be summarized in Table 2. In short, for the system involving SNAP25, Syntaxin and VAMP2, we should only consider two types of initial conditions: If the initial conditions only include initial concentrations of SNAP25, Syntaxin and VAMP2, but no tSNARE, the final steady state of fusion 
is equal to the least initial concentration of SNAP25, Syntaxin and VAMP2. In the case of non-zero initial concentration of tSNARE (
), however, the final steady state 
can be much higher as long as the initial concentration of VAMP2 (vSNARE) is sufficiently large. This case is particularly interested because in vivo, SNAP25 and Syntaxins may have been already preincubation (preformed) into tSNAREs on the plasmic membrane, before vSNARE proteins (VAMP2 in our case) approach, as carried by vesicles.
Table 2. A brief summary for the stabilizing analysis of the fundamental subsystems without regulation.
| Initial condition for the first reaction | Initial condition for the second reaction | Fusion level at steady state, [FHC] |
|
|
|
|
|
|
|
|
|
|
|
|
Our analysis explains why the outcome of fusion process depends on the way to put these three proteins into the system [21]. One is the sequential process: SNAP25, Syntaxin and VAMP2 proteins are added into the system in order such that virtually no tSNARE protein complex has been formed when the reaction begins. The other one is the preformed process: After SNAP25 and Syntaxin proteins have been preincubation (preformed) into tSNARE, VAMP2 proteins are then added to initiate the fusion reaction. Numerical simulations have shown that the preformed process reaches the steady state much faster than the sequential one (Fig.3), which is consistent with in vitro experimental data (the embedded panel) [21].
Figure 3. A comparison between proformed and sequential fusion processes.
Numerical simulation results are presented, whereas the experimental results are in the embedded plot from [21].
Stability Analysis on MUNC18-dependent Regulation
We furthermore study the stability behavior of the system involving the regulatory protein MUNC18. As discussed above, the MUNC18-dependent regulation has two types: (
) It binds tightly to a closed conformation of sytanxin that precludes the syntaxin’s involvement in the fusion process, suggesting that MUNC18 inhibits fusion by regulating the formation of tSNARE. And (
) it can assemble with SNARE complexes (FHC) to accelerate membrane fusion in late stages when the concentration of four helical bundles (FHC) is high enough.
We make the following assumptions to simplify the subsystem with MUNC18-dependent regulation. Considering the situation that tSNARE has been preformed and reaction of SNAP25, sytanxin and tSNARE has reached the equilibrium, we claim that the function of MUNC18 can be characterized as follows.
 |
where FHC
is the complex of MUNC18 and FHC, and it behaves similar to FHC to help the fusion process; and
 |
where 
stands for the binding rate of MUNC18 onto the syntaxin in closed conformation. In the above reactions, the concentrations of four helical bundles, FHC, and four helical bundles binding with MUNC18, FHC** reflect the level of fusion. It has been shown that the disassociation rate of MUNC18-syntaxin complex is very small comparing to the binding rate, so that the second reaction is considered as an irreversible one.
Introducing variables 
, 
, 
, 
, and 
, the subsystem involving MUNC18 is rewritten as.
 |
 |
 |
 |
 |
(11) |
Using the mathematical approaches similar to the case of no regulation, we have studied the stability of system Eq.(11). Assume the binding rate 
, reaction rates 
are in the range given by the reference [17]–[19], we have shown the existence of steady states of Eq.(11), including bi-stability. As the result has been rigorously presented in the section of Data and Methods, we are mainly interested in the final fusion level, as measured by 
. Under the assumption that the initial concentration of FHC** is zero, i.e., 
, we interpret our results as follows.
(
) If 
, there exist two bi-stable states for the final fusion levels: One is the high fusion level, which is given by.
 |
(12) |
In this case, at the steady state, the concentrations of free MUNC18 and free vSNARE (VAMP2) are virtually zero, which mean all of these proteins exist in the form of FHC and/or FHC**. The second steady-state is the low fusion level (
), the up-bound of 
is actually 
, whereas the low-bound is given by.
 |
(13) |
On the other hand, the low steady state fusion level, 
is somewhere between 
and 
. In this case, the steady state, the concentrations of free MUNC18 and free tSNARE are virtually zero, which mean all of these proteins exist in the forms of FHC and FHC**.
(
) Otherwise, there exists only one steady state that is locally stable, and the final fusion 
is somewhere between 
.
Hence, with the regulation of MUNC18, the steady states of final level of fusion is controlled by the initial concentration of MUNC18: The behavior of bi-stability exists only when the initial concentration of MUNC18 is intermediate, whereas the boundary is determined by initial concentrations of tSNARE, VAMP2 and FHC. A lower or higher 
results in a single steady state of the final fusion level. Moreover, the final fusion level depends on 
, suggesting that the preincubation (preform) of tSNARE is an important factor. Indeed, using numerical simulations, we have shown that for the system involving SNARE proteins, complexes and MUNC18, preformed assays have two advantages over the sequential one: first, preincubation advances reaction rates; second, preincubation support more fusion than sequential assays. Finally, we comment that the regulation mechanism of MUNC18 may be threshold dependent, i.e. there exists an optimal threshold ? which depends on the initial concentration of tSNARE and four helical bundle only, such that MUNC18’s regulation function during fusion is maximized when the initial concentration of MUNC18 reaches the threshold (Xia et al, unpublished results).
Conclusive Remarks
In this study, we present a framework for modeling protein interaction network which are involved exocytotic process. The framework is based on classic chemical kinetic model that generates insights into system dynamics and stability. The computational experiments and mathematical analysis reveal that the frame reconstruct biological experimental observation successfully and is able to provide useful predictions.
Methods
Simulation Procedures
The kinetics simulation and analysis of the whole system or the subsystems were implemented in Matlab7.0R. Differential equations were solved using the ODE23s routine. For testing the robustness of parameters, we generated 2000 random parameter sets using Latin Hypercube Sampling when all parameters are varied 
% relative to their original values, with a a uniform distribution for each parameter.
The concentrations of reactant proteins are given in molar units. For non-soluble proteins such as vSNARE and VAMP2, we followed the work in [10] and based the protein concentration estimation on the concentration of secretory vesicles in molar. During the exocytotic process, the size of vesicle pools varies with respect to different cell types from 200 to 3000. Hence, the molar concentration of vesicles was estimated in the range of 0.2–30 
. Accordingly, the concentration of VAMP2 is considered to be in an identical range of vesicle concentration (0.2–30 
) [10]. The tSNARE proteins such as SNAP25 and syntaxin are thought to be vastly expressed in vivo and the studies [1]–[6] evaluated the concentration of these protein in a range of 0.1–100 
. The essential regulatory protein Munc18 is known to be expressed at much lower levels, compared to SNARE proteins, with the concentrations in range of 1–30 
[3]–[4], [10].
Algorithm for the Estimation of Rate Parameters
To recover the appropriate reaction rates, we apply technique of solving the inverse problem introduced by [27] to Eq.(10). Some useful results are presented below. To be concise, the ODE system Eq.(10) is written as 
, 
is the initial conditions, and the parameter set 
.
The inverse problem claims that the parameter identification of Eq.(10) is equivalent to the optimization problem of.
 |
(14) |
subject to 
and 
, where 
is the parameter space in 
and 
is regularized energy functional
 |
(15) |
where 
and 
are Tikhonov regularization parameter [25], 
is parameter-data mapping, 
is experimental data, and 
is the penalty function to guarantee the positivity of reaction rates.
The rational of the inverse problem is based on the following theorem: Suppose the solution of Eq.(10) 
is smooth, where 
is the observation time. Then, given observed data on each time point in 
, the parameters identified by the inverse problem are locally unique with respect to the initial condition. Under the assumption that all of the reaction rates are roughly constant, the optimization problem is solved through a gradient-based method. The brief algorithm is sketched below:
Given initial condition
, solving ODE system 
by fourth order RK and mapping it on the observation data set,Gradient representation: using forward difference to approximate
,Applying steepest descent to approach the global minimum starting with some initial guess,
Using adjoint scheme to approximate Hessian
of 
,Using the approximate solution given by step 2 as initial guess , and using Quasi-Newton method with
to find the appropriate parameter.
Stabilizing Analysis of the Fundamental Subsystem
The formal claim from the stabilizing analysis of the fundamental subsystem Eq.(10) is as follows: Define 
, and 
and 
. Let 
(
) and 
(
) be the steady-states for 
, 
, 
, 
, and 
, respectively, and the steady-state vector 
.
Case-
Assume the initial concentrations of SNAP25 and syntaxin are the same so that 
. Denote 
, provided 
. There are two stable steady states, 
and 
: (
) If 
, we have.
 |
(16) |
and (
) otherwise
 |
If 
, the reduced steady state is 
, which is locally stable.
Case-
Consider the case of 
. Provided 
where 
, there are four steady states:
if
and 
, the steady state is 
and it is stable locally;if
and 
, the steady state is 
and it is a stable node locally;if
and 
, the steady state is 
and it is stable locally;if
, 
, and 
, the steady state is 
and it is a stable node locally.
Case-C
A more general case is that the reaction ratios (
, 
) and concentrations of SNARE proteins and complexes are in the same order, i.e., 
and 
. The steady states are.
 |
 |
 |
(18) |
and if 
 |
(19) |
where
 |
(20) |
Note that 
and 
are locally stable nodes. When 
is small sufficiently comparing to the concentrations of SNARE proteins and complexes, 
is reduced to.
 |
 |
(21) |
Proof. A concise proof is presented below. From the definition of 
to 
, straightforward calculation simplifies the fundamental subsystem Eq.(10) as follows.
 |
 |
(22) |
For Case-A that 
, Eq.(22) can be further simplified to be.
 |
 |
(23) |
Notice that reaction rates recovered from the experimental data imply 
, so that compared to the concentrations of SNARE complexes, 
is negligible. Thus, the 
-nullcline determined by Eq.(23) so that.
 |
Denote 
, 
-nullcline is given by.
 |
The steady states are yielded by intersecting the nullclines, and the biological interesting steady states are therefore given by 
and 
. Straightforward calculation implies the steady states 
and 
are a pair of opposite vertexes, and the relationship of 
and 
determines the choice of these two steady states. If 
, the only possible steady state is 
; if 
, the only possible steady state is 
.
To investigate the stability of those steady states, we calculated the corresponding Jacobian for system Eq.(23) and then evaluate the two eigenvalues, denoted by 
and 
, respectively. For steady state 
, two eigenvalues for the Jacobian have no zero real part because of 
, so that steady state 
is a hyperbolic point of system Eq.(23). By Hartman-Grobman theorem, there exists a homeomorphism mapping the trajectories of Eq.(23) in an open set containing 
onto trajectories of its linearized system in an open set containing 
. Furthermore, the homeomorphism preserves the parameterizations by time. Therefore, local behaviors of 
is characterized by its corresponding Jacobian, leading to 
and 
. Therefore, steady state 
is stable locally.
For steady state 
, we calculated the corresponding Jacobian and showed that none of the eigenvalues has zero real part, so that 
is a hyperbolic steady point of system Eq.(23). The local behavior of trajectories of (Eq.(23) in the neighborhood of 
is characterized by its linearized system with respect to 
. As the trace of correspondiong Jacobian is less than zero, we show 
is stable locally.
In the same manner, we have shown the results presented in case-B and case-C.
Stabilizing Analysis of MUNC-18 Regulation
For the system described in Eq.(17), there are three steady states, which are.if 
and thus 
,then
 |
(24) |
for any 
, or
 |
(25) |
for any 
.if 
, then
 |
(26) |
for any 
.where 
and 
. These steady states are locally stable.
Proof: The proof is similar to the case of system Eq.(10) without regulation.
Acknowledgments
We would like to thank Yeon-Kyun Shin, Jian-Song Tong, and Julia Dickerson for their valuable suggestions and comments.
Footnotes
Competing Interests: The authors have declared that no competing interests exist.
Funding: The authors have no support or funding to report.
References
- 1.Weber T, Zemelman BV, McNew JA, Westermann B, Gmachl M, et al. SNAREpins: Minimal Machinery for Membrane Fusion. Cell. 1998;92:759–772. doi: 10.1016/s0092-8674(00)81404-x. [DOI] [PubMed] [Google Scholar]
- 2.Sollner T, Bennett MK, Whiteheart SW, Scheller R, Rothman JA. Protein assembly-disassembly pathway in vitro that may correspond to sequential steps of synaptic vesicle docking, activatgion, and fusion. Cell. 1993;75:409–418. doi: 10.1016/0092-8674(93)90376-2. [DOI] [PubMed] [Google Scholar]
- 3.Chen YA, Scheller RH. SNARE-mediated membrane fusion. Nat Rev Mol Cell Biol. 2001;2:98–106. doi: 10.1038/35052017. [DOI] [PubMed] [Google Scholar]
- 4.Burgoyne RD, A. Morgan A. Secretary Granule Exocytosis. Physiol Rev. 2003;83:581–632. doi: 10.1152/physrev.00031.2002. [DOI] [PubMed] [Google Scholar]
- 5.Lin RC, Scheller RH. Mechanisms of synaptic vesicle exocytosis. Annu Rev Cell Dev Biol. 2000;16:19–49. doi: 10.1146/annurev.cellbio.16.1.19. [DOI] [PubMed] [Google Scholar]
- 6.Burgoyne RD, Morgan A. Membrane trafficking: three steps to fusion. Curr Biol. 2007;17:255–258. doi: 10.1016/j.cub.2007.02.006. [DOI] [PubMed] [Google Scholar]
- 7.Heinemann C, Ruden LV, Chow RH, Neher E. A two-step model of secretion control in neuroendocrine cells. Pflügers Arch. 1993;424:105–112. doi: 10.1007/BF00374600. [DOI] [PubMed] [Google Scholar]
- 8.Jahn R, Lang T, Sudhof TC. Membrane fusion. Cell. 2003;112:519–533. doi: 10.1016/s0092-8674(03)00112-0. [DOI] [PubMed] [Google Scholar]
- 9.Lu XB, Zhang F, McNew JA, Shin YK. Membrane fusion induced by neuronal SNAREs transits through hemifusion. J Biol Chem. 2005;280:30538–30541. doi: 10.1074/jbc.M506862200. [DOI] [PubMed] [Google Scholar]
- 10.Mezer A, Nachliel E, Gutman M, Ashery U. A new platform to study the molecular mechanisms of exocytosis. J Neurosci. 2004;24:8838–8846. doi: 10.1523/JNEUROSCI.2815-04.2004. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 11.Moser T, Neher E. “Estimation of mean exocytic vesicle capacitance in mouse adrenal chromaffin cells,”. Proc Natl Acad Sci USA. 1997;94:6735–6740. doi: 10.1073/pnas.94.13.6735. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 12.Misura KMS, Schelle RH, Weis WI. Self-association of the h3 region of syntaxin 1a implications for intermediates in snare complex assembly. J Biol Chem. 2001;276:13273–13282. doi: 10.1074/jbc.M009636200. [DOI] [PubMed] [Google Scholar]
- 13.Sudhof TC. The synaptic vesicle cycle. Annu Rev Neurosci. 2004;27:509–547. doi: 10.1146/annurev.neuro.26.041002.131412. [DOI] [PubMed] [Google Scholar]
- 14.Brose N, Petrenko AG, Sudhof TC, Jahn R. Synaptotagmin: a calcium sensor on the synaptic vesicle surface. Science. 1992;256:1021–1025. doi: 10.1126/science.1589771. [DOI] [PubMed] [Google Scholar]
-
15.Chapman ER. Synaptotagmin: a Ca
sensor that triggers exocytosis? Nat Rev Mol Cell Biol. 2002;3:498–508. doi: 10.1038/nrm855. [DOI] [PubMed] [Google Scholar] -
16.Bhalla A, Chicka MC, Tucker WC, Chapman ER.
-synaptotagmin directly regulates t-SNARE function during reconstituted membrane fusion. Nature Struc Mol Biol. 2006;13:323–330. doi: 10.1038/nsmb1076. [DOI] [PubMed] [Google Scholar] - 17.Ciufo LF, Barclay JW, Burgoyne RD, Morgan A. Munc18–1 regulates early and late stages of exocytosis via syntaxin-independent protein interactions. Mol Biol Cell. 2005;16:470–482. doi: 10.1091/mbc.E04-08-0685. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 18.Margittai M, Widengren J, Schweinberger E, Schroder GF, Felekyan S, et al. Single-molecule fluroscence resonance energy transfer reveals a dynamic equilibrium between closed an dopen conformations of syntaxin I. Proc Natl Acad Sci USA. 2003;100:15516–15521. doi: 10.1073/pnas.2331232100. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 19.Fasshauer D, Margittai M. A transient N-terminal interaction of SNAP-25 and syntaxin nucleates SNARE assembly. J Biol Chem. 2004;279:7613–7621. doi: 10.1074/jbc.M312064200. [DOI] [PubMed] [Google Scholar]
- 20.Giraudo CG, Eng WS, Melia TJ, Rothman JE. A clamping mechanism involved in SNARE-dependent exocytosis. Science. 2006;313:676–80. doi: 10.1126/science.1129450. [DOI] [PubMed] [Google Scholar]
- 21.Pobbati AV, Stein A, Fasshauer D. N- to C-terminal SNARE complex assembly promotes rapid membrane fusion. Science. 2006;313:673–676. doi: 10.1126/science.1129486. [DOI] [PubMed] [Google Scholar]
- 22.Shen JS, Tareste DC, Paumet F, Rothman JE, Melia TJ. Selective activation of cognate SNAREpins by SEC1/MUNC18 proteins. Cell. 2007;128:183–195. doi: 10.1016/j.cell.2006.12.016. [DOI] [PubMed] [Google Scholar]
- 23.Weninger K, Bowen ME, Chu S, Brunger AT. Singel-molecule studies of SNARE complex assembly reveal parallel and antiparallel configurations. Proc Natl Acad Sci USA. 2003;100:14800–14805. doi: 10.1073/pnas.2036428100. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 24.Keener J, Sneyd J. Mathematical Physiology, New York Springer-Verlag. 2001.
- 25.Lorenzi A. An introduction to identification problems via functional analysis, Utrecht: VSP-BV. 2001.
- 26.Perko L. Differential equations and dynamical systems, New YorkL Springer-Verlag. 2000.
- 27.Vogel CR. Computational Methods for inverse problem, SIAM, 2002. 2002.
- 28.de Vries G, Hillen T, Lewis M, Muller J, Schonfisch B. A course in mathematical biology-quantitative modeling with mathematical and computational methods, SIAM Philadelphia. 2006.
- 29.Xia T, Tong J, Rathore SS; Gu, X, Dickerson J. Comparative network motif design rationalizes Sec1/Munc18-SNARE regulation mechanism in exocytosis. BMC Systems Biology. 2012. (in press). [DOI] [PMC free article] [PubMed]