Computer Mathematical Simulation of the Dynamic Interactions

Abstract

The regulation of neurotransmitter release has long been an important area in neurochemistry, affecting such facets of brain and nerve function as cognition, emotion, memory, and metacognitive processing. To this end, in vitro studies of indepdendent components of this system have been studied and characterized. Kinetic constants obtained through previous studies were used in this work to construct a novel theoretical model for the pre_synaptic neurotransmitter release.

Synapsins constitute a group of three types of phosphoproteins that interact with synaptic vesicles and are a vital component in the regulation of neurotransmitter release. By acting as a cytoskeletal "tether," they are able to mediate the translocation of synaptic vesicles containing neurotransmitter within a cell. Their affinity for actin filaments and synaptic vesicles is modulated by their phosphorylation thereby allowing the dynamic modelling of the interaction of these components.

A computer mathematical model was proposed to illustrate the interactions between Synapsin (phosphorylated and dephosphorylated forms), actin filaments of the cytoskeleton, and synaptic vesicles. Literature constants were used to approximate the in vivo interactions between the four components. Attempts were made to formulate a related system to integrate calcium/calmodulin_dependent Kinase (CaM Kinase II) and its interactions with Synapsin Ia.

Introduction

Synapsins represent a family of abundant synaptic vesicle-associated phosphoproteins which are involved in synaptic vesicle dynamics in nerve terminals (Hosaka and Sudhof, 1998; Geppert et al ., 1994; Sudhof, 1995). They account for approximately 9% of the membrane proteins associated with synaptic vesicles (Esser et al ., 1998). Synapsin were originally discovered in 1977 by Ueda and Greengard and were originally called “Protein 1.” Until recently, two synapsin genes were known: Synapsin I and Synapsin II (Sudhof, 1995). The products of these two genes are alternatively spliced to produce the four synapsin proteins: synapsins Ia, Ib, IIa, and IIb (Sudhof, 1995). Hosaka and Sudhof (1998) recently, however, proposed the existence of a novel Synapsin–Synapsin III.

Thus far, synapsin interactions with brain spectrin, actin filaments, neurofilaments, and microtubules have been reported (Bennett et al ., 1991). Virtually all neurons, regardless of the neurotransmitter released, contain synapsins (Greengard et al ., 1994). In nerve terminals, Synapsin I is specifically located on the cytoplasmic surface of small synaptic vesicles (Benfenati et al ., 1989). There is strong evidence that synapsin I is a regulator of neurotransmitter release through the immobilization of these small synaptic vesicles to the cyotoskeleton, thereby allowing the accumulation of a releasable pool of neurotransmitter (Bahler et al ., 1989). The maintenance of this releasable pool of neurotransmitter allows the rapid recovery and firing of neurons. Further, Bahler et al . (1989) proposed that this regulation is strongly dependent upon the phosphorylation of the synapsin I molecule itself.

In the rat, Synapsins Ia and Ib are 704- and 668-amino acid proteins and Synapsins IIa and IIb are 586- and 479-amino acid proteins, respectively (Greengard et al ., 1994). Between rat, bovine, and human forms, Synapsin maintains a high degree of identity (Esser et al ., 1998). In each case, it comprises five domains: A, B, C, D, and E/F. Each domain is distinguished by its composition and conservation. Of these, Esser et al . (1998) point out that the C-domain is the most highly conserved. It is 39% hydrophobic and 27% charged, and is located in the amino terminal, globular region of the protein.

Synapsin I has a collagenase-insensitive head domain and a collagenase-sensitive tail region (Benfenati et alI., 1989). It has three recognized sites at which it may be phosphorylated by cAMP-dependent protein kinase or Ca 2+ /calmodulin-dependent protein kinase (CaMKII). Benfenati et al . (1989) discovered one serine residue in the head domain (“Site 1"), which is phosphorylated by cAMP-dependent protein kinase, and two serine residues in the tail domain, phosphorylated by CaMKII. It has been shown that phosphorylation at these residues influences the affinity of synapsin for synaptic vesicles (Benfenati et al ., 1989; Esser et al ., 1998; Benfenati et al ., 1989).

Neurotransmitters are stored in synaptic vesicles and released by exocytosis. The only known function of these small synaptic vesicles is the storage and release of neurotransmitters. This relative simplicity has led to the synaptic vesicle being one of the best-studied organelles in biology. Although synapsin I is a very well-characterized protein, and many equilibrium interactions have been demonstrated, as yet there have been no attempts at simulating the real-time interactions between synapsin I, synaptic vesicles, and the cytoskeleton. The purpose of this project is therefore to construct a real-time mathematical computer simulation that reflects possible interactions between phosphorylated and dephosphorylated synapsin I with synaptic vesicles and actin filaments using estimated kinetic constants and computer simulation software.

There is a general consensus that synapsins are necessary for regulation of synaptic vesicle exocytosis and therefore neurotransmitter release. In spite of this vital regulatory function, several experimenters have proven that the lack of synapsins is not lethal. Experiments on mice have led to more debate regarding the necessity of synapsins to normal neurological function. Certain experiments have revealed that mice lacking synapsins expressed no gross anatomical abnormalities, yet experienced grand mal seizures proportional to the number of mutant synapsin alleles (Rosahl et al ., 1995) and others revealed that mice lacking synapsin I exhibited no apparent changes in well-being or gross nervous system function. This evidence suggests that synapsin I normally functions to inhibit neurotransmitter release on a millisecond timescale immediately following a calcium signal, generally thought to occur through C-terminal phosphorylation of synapsin by CaM Kinase II (Sudhof, 1995).

Benfenati et al . (1989) used hydrophobic photolabelling to study the interactions between synaptic vesicles and synapsin I. The synapsin-synaptic vesicle complexes were preserved by solubilizing and reconstituting in phosphatidyl choline.

There is evidence that the head of the protein is involved in the binding of the phospholipid bilayer while the tail binds a separate vesicle protein, which is strongly modulated by phosphorylation events and ionic strength (Benfenati et al ., 1989; HUttner et al ., 1993). This implies two types of interactions between synapsin and synaptic vesicles. One interaction, purely hydrophobic in nature, does not seem to be influenced by phosphorylation state or ionic strength. Another interaction, due to protein-protein interactions in the tail region, allows regulation of interactions by changes in phosphorylation state or ionic strength of the environment. The proposed computer simulation would incorporate the phosphorylation state of the medium, but not ionic strength or hydrophobic factors into the simulation.

The main focus of the synaptic vesicle cycle is exocytosis by membrane fusion. The study of the synaptic vesicle cycle is vitally important to the study of neurotransmitter release and design of neural networks.

Although the synaptic environment imposes special requirements on signalling, such as localized and rapid signals which can be repeated at high frequencies, both up- and down-regulated, the process of exocytosis employed by synaptic vesicles in man is identical to the process used by yeasts. Sudhof (1995) described four main methods to study the synaptic vesicle cycle:

The synaptic vesicle cycle takes approximately one minute to complete. Of this, the actual exocytosis event takes approximately one third of a second. The endocytotic event requires approximately 5 seconds. Sudhof (1995) describes nine distinct steps which comprise the synaptic vesicle cycle:

Proposed Kinetic Simulation of Synapsin I Interactions

Synapsin I interactions with actin, ATP, and synaptic vesicles are well documented (Sudhof, 1995; Hosaka and Sudhof, 1998; Bennet et al ., 1991; and Esser et al ., 1998). To date, however, a mathematical simulation of the kinetic interactions of all four players concurrently has not been published. Using literature kinetic and affinity constants for both phosphorylated and dephosphorylated Synapsin I interactions with actin and synaptic vesicles, a pre-equilibrium computer-based mathematical simulation of these interactions can be constructed. In order to do this, the kinetics between each of the players is characterized independently, and integrated as the sum of parallel processes using a computer-based solving engine.

Synapsin I Interactions with Small Synaptic Vesicles

There are two main aspects to interactions between Synapsin I and small synaptic vesicles (SV): hydrophobic binding of the head region to the phospholipid membrane and binding of the tail region to a non-lipid component of the membrane (Schiebler et al ., 1986; Benfenati et al ., 1989). Studies by Benfenati et al . (1989) and Schiebler et al . (1986) showed that binding of the hydrophobic head group to the membrane is largely the same whether interacting with native synaptic vesicles or mixed liposomes mimicking synaptic vesicles.

As previously noted, tail binding to synaptic vesicles is strongly affected both by phosphorylation of sites 2 and 3, and by ionic environment, implying a protein-protein interaction more comlpex than the simple hydrophobic interaction of the head group. Benfenati et al . (1989) reported a differential binding affinity for synaptic vesicles according to phosphorylation of sites 1, 2, and 3 and ionic environment. Dephosphorylated synapsin-bound synaptic vesicles with high affinity and saturability, exhibiting K D =10 mM at 40 mM NaCl and B max =800 fmol/:g of synaptic vesicles. This affinity could be decreased substantially by phosphorylation of the tail domain serine residues or by increasing the ionic strength of the mixture beyond 40 mM. Synapsin I phosphorylated at site I did not show a high degree of difference to dephosphorylated synapsin in experiments with both native synaptic vesicles and artificial phospholipid bilayers. This is consistent with the theory that hydrophobic interactions are largely responsible for the hydrophobic head domain of Synapsin I. A single phosphate group is not likely to supply either enough charge differential or steric hindrance to disrupt the strong hydrophobic interactions between the rest of the head domain and the phospholipid membranes. There is a difference, however, in the results when considering phosphorylation at sites 2 and 3 in the tail domain, which is not known to interaction significant through hydrophobic interactions with membranes. The tail domain is also known to be involved in interactions with non-lipid components in native synaptic vesicles. These components may not be present in artificial phospholipid bilayers which may explain why there is little difference in binding affinity for synapsin I phosphorylated at sites 2 and/or 3 compared to dephosphorylated synapsin or site 1-phosphorylated synapsin. In experiments with native synaptic vesicles, however, there is a six-fold increase in dissociation constants when the tail region is phosphorylated. In non-hydrophobic interactions with non-lipid components, the steric hindrance or charge differential caused by the addition of the phosphate groups may explain the six-fold increase in dissociation constants relative to the non-phosphorylated synapsin-vesicle complex.

In considering other factors affecting non-hydrophobic binding, pH and ionic strength support the theory that protein-protein interactions are responsible for the tail binding. According to Schiebler et al . (1986), Synapsin I-vesicle binding was weakened by pH # 5 and binding was not observed at pH less than 3. This indicates either a partial denaturation of the protein, steric profile differences in protonated versus deprotonated amino acids, or interference with charge interactions between the tail region and vesicle. Additionally, when bovine Synapsin I was extracted at pH less than 3, affinities two- to three-fold lower than detergent-salt-extracted synapsin I were observed (Schiebler et al ., 1986).

Actin Nucleation and Elongation

There is evidence that the regulatory function synapsins play in the release of neurotransmitters through exocytosis is accomplished through the tethering of synaptic vesicles to cytoskeletal elements (Bahler et al ., 1989). The study and analysis of interactions between actin microfilaments and Synapsin I provides valuable insight into the synaptic vesicle cycle and neurotransmitter regulation. Normally, actin monomers will spontaneously polymerize when it has fulfilled three requirements: 1. the preparation has reached a certain critical concentration of G-actin, 2. there is a nucleation “seed” oligomer of F-actin, and 3. elevated concentrations of potassium and magnesium are present (Fesce et al , 1992). In 1992, Fesce et al . demonstrated that Synapsin I was able to form Synapsin-G actin complexes which caused actin to polymerize where spontaneous polymerization is neglibile; furthermore, this interaction is strongly affected by phosphorylation and dephosphorylation of synapsin.

K D (nM)

168

No binding observed

Actin polymerization rate is dependent upon a number of factors. These include number of pre-existing filaments, the concentration of monomeric actin, and the rates at which monomers are added to the barbed end of the filament and removed from the pointed end (Fesce et al ., 1992).

The relationship between Synapsin I and G-actin, however, proves much more complex. According to Fesce et al . (1992), the interactions do not follow simple stoichiometric ligand-receptor kinetics; in fact, the concentration of monoeric G-actin itself influences the interactions. When Synapsin I is added to actin preparations not in conditions normally amenable to spontaneous polymerization, is is able to trigger the formation of actin filaments, likely due to the formation of synapsin I- G-actin pseudonuclei (Fesce et al ., 1992). Interactions between filamentous actin, however, and Synapsin I have not yet been investigated and so the precise rate constants are unknown between filamentous actin and Synapsin I. For the purposes of this computer model, they have been estimated. Additional complications may be introduced with modulation of Synapsin I-actin polymerization by ionic strength and Synapsin phosphorylation. Synapsin I affects the microfilament polymerization rate. For the purposes of this computer model, it was determined that as a simplifying assumption, the interactions of Synapsin I with actin are with filamentous actin; monomeric actin is not considered.

Interactions with the Cytoskeleton

By virtue of its interactions with microtubules, Synapsin I is a member of the membrane-bound microtubule-associated proteins (Bennett et al ., 1991). Baines and Bennett (1986) found that it co-isolates with microtubules from both whole-brain extracts put through polymerization and depolymerization cycles, as well as purified tubulin preparations. To demonstrate this, they pelletted taxol-stabilized microtubules prepared from phosphocellulose-purified bull brain. The pellets were then analyzed by SDS-PAGE. The Synapsin I head and tail fragments consistently co-sedimented with microtubules, the tail being more common than the head. The experimenters, however, were not able to extrapolate specific association constant values from this experiment. These results seem surprising considering that sequence comparisons between Synapsin I and the microtubule-binding bovine tau protein revealed no significant sequence identities or resemblance to MAP 1b microtubule binding domain motif “Lys-Lys-Glu-Glu.” Studies of the tubulin-binding portions of Synapsin may yield valuable information about its actin-binding activity; Correas et al . (1990) found that the microtubule-associated protein 2 (MAP 2) tubulin binding sites also had significant actin-binding activity. Immunoelectron microscopy also indicated a relationship between the tubulin-binding sites of tau protein and actin-binding activity. The microtubule- and microfilament-binding sites of Synapsin I may be one and the same (or at least shared). This, however, could imply a competitive relationship between actin and tubulin binding. For purposes of this simulation, it is assumed that interactions at potential microfilament/microtubule binding sites are with actin filaments only.

Software Simulation of Interactions

The simulation of real-time interactions between phosphorylated and dephosphorylated Synapsin I, synaptic vesicles, and actin microfilaments will be simplified to consider the affinities for these four components only. Because a change in concentration of any one of the components implies a change in all of the others, the changes in concentration for each of the four molecules must be simultaneously solved. Facilitating this feat, the SCoP software program will allow the tedious and insurmountable task of manual calculations to be completed by computer. As per the kinetic reaction web illustrated below, each potential reaction will be represented by an equation for which a forward and backward reaction are characterized by a discrete constant. For example, the reaction of dephosporylated synapsin binding to synaptic vesicles is represented by:

S + V W SV

The forward reaction is characterized by a k on constant, and the reverse by a k off . The software program is able to calculate the concentration of bound synaptic vesicles as by the following relationship:

Similarly, k cat1 and k cat2 are used to calculate the population of phosphorylated versus non-phosphorylated synapsin.

Materials and Methods

Real-time computermathematical modeling was accomplished using Simulation Control Program (Simulation Resources, Inc.; Redland CA ). The programming involved writing a MOD file using a text editor and compiling this MOD script using MAKESCOP. The end result is an executable DOS file, which can be used to run the simulation. Output was taken in the form of a tabulated DOS text file, which was imported into Microsoft Excel97 and plotted as a graph.

Kinetic constants used for simulation were estimated from literature constants. In order to vary the constants used in each run of the program, separate “VAR” files were created according to each condition. The “VAR” file contents were constructed for each condition as follows:

In order to simulate a higher population of phosphorylated versus non-phosphorylated Synapsin, the kcat values were modified to favour the phosphorylated forms.

Results

Output from the SCoP software consisted of a graphical representation of the amount of synaptic vesicles bound to the cytoskeleton versus time.

Figure 1 illustrates the proportion of synaptic vesicles bound to actin filaments by synapsin. It is expressed as a proportion of the starting concentration of synapsin in the mixture, 1 x 10-6.

Figure 2 illustrates the proportion of synaptic vesicles bound to actin filaments by synapsin when a large population is phosphorylated. In vivo , this would be effected by phosphorylation at sites 2, 3 by calcium/calmodulin-dependent protein kinase. In the simulation, it was simulated by changing the two constants representing the phosphorylation of synapsin to favour the phosphorylated form.

Discussion

Synapsin Ia represents one member of a family of five types of neuron-specific phosphoproteins. It is believed that its role in regulation of neurotransmitter release by synaptic vesicle fusion and exocytosis in the presynaptic nerve terminal involves the differential association and dissociation with neurotransmitter-carrying synaptic vesicles and elements of the cytoskeleton (Bahler and Greengard, 1987; Stefani G et al ., 1997). Because of the high frequencies with which action potentials may occur, it is necessary that a releasable pool of neurotransmitter be maintained and cycled quickly. The elucidation of the mechanisms responsible for these interactions is therefore critical in the understanding of the processes governing complex psychological processes, such as memory, cognition, emotion, and psychological disorders.

Many studies to date have involved equilibrium analysis of individual interactions between Synapsin proteins and either synaptic vesicles or cytoskeletal elements, but not as a whole. The value of a computer-based real-time simulation lies in the flexibility and ease of changing controlled variables, and especially in the reductionist approach to simplifying a problem to its most critical components. The use of Pareto analysis is therefore a valuable tool in the selection of the most critical factors in a relationship the modulation of which factors yields the greatest variability. The direct observation of the pre-equilibrium kinetics of the interactions between Synapsin and both actin filaments and synaptic vesicles has proven to be difficult, and although pre-equilibrium rate constants have been obtained for various aspects of this relationship through such techniques as flourescence resonance energy transfer (Stefani G, et al ., 1997), some critical aspects remain to be characterized. For example, Stefani et al . (1997) found that it was not possible to obtain the specific kinetic kon and k off rates for synapsins and synaptic vesicles using traditional methods involving exogenously-added activated phosphorylating calcium/calmodulin kinase; fluorescence transfer in the case of phosphorylation at the C-terminal sites by CaMKII greatly reduced binding of synapsin to synaptic vesicles, and the increase in ionic strength meant to cause dissociation did not cause appreciable change to the already-low levels of bound synaptic vesicles.

Although Synapsins have been shown to have significant binding relationships with actin filaments, neurofilaments, and microtubules, the major component believed to be involved in the storage of synaptic vesicles are actin filaments (Bahler and Greengard, 1987) because previous studies had shown that neither neurofilaments nor microtubules are present in appreciable amounts in the region of the presynaptic nerve terminal where synaptic vesicles are generally located (Roots, 1983).

Synaptic vesicles have been shown to have two types of interactions with synapsin proteins. The hydrophobic interaction between the head portion of the synapsin molecule and the synaptic vesicles was demonstrated to be strictly hydrophobic in nature, and affected by neither ionic strength nor phosphorylation state. A second interaction, more relevant to regulatory function, is the protein-protein interaction between the tail portion of synapsin and a protein component of synaptic vesicles. Stefani et al . (1997) reported that the dissociation constants for mixed phospholipid vesicles versus actual synaptic vesicles was higher, implying that there is a specific component in synaptic vesicles which allows for greater binding potential. The effects of phosphorylation of synapsin in the simulation was therefore designed with site 2/3 phosphorylation as a target, implying further work to incorporate CaM Kinase II regulation into the model.

Figure 1 illustrates the simulation using the default parameters derived both from literature values and estimated values. The rapid binding and high saturability exhibited by the fluorescence resonance energy transfer experiments (Stefani et al ., 1997) are also exhibited by the computer simulation.

Literature values for the rate constants governing the association of both the phosphorylated and dephosphorylated forms of synapsin for synaptic vesicles were obtained from Stefani et al . (1997). Literature had previously existed which described the equilibrium relations between synapsin and actin filaments, but as yet, no pre-equilibrium binding kinetics data are available. It is possible that using the same FRET technology used to determine the rate constants for the synapsin-vesicle binding this constant could be determined in future studies.

It has been demonstrated that the affinity of synapsin for synaptic vesicles can be based on the state of phosphorylation of sites 2 and 3 in the tail region and is irrespective of the phpshorylation state of site 1 in the head region (Stefani et al ., 1997). For this reason, the modelling of the phosphorylation state assumes phosphorylation of sites 2 and 3 by calcium/calmodulin-dependent protein kinase II (CaM KII) and not site 1 by protein kinase A (PKA). This is represented in the model by a pair of static constant, k cat1 and k cat2 . In an in vivo system, these two constants would be supplanted by the specific activity of the CaMKII present in the system. Future direction would include the incorporation of a CaMKII model which would, in turn, replace the static constants used in this model.

In order to take into account the phosphorylation of the tail sites by CaMKII, the factors affecting the activity of CaMKII must be analyzed. Studies have shown that CaMKII is a multisubstrate kinase which is able to phosphorylate any of a given set of substrates in the presence of calcium/calmodulin and given a phosphate source, ATP (Stefani et al ., 1997; Hanson et al ., 1994). The actual implementation of such a model, however, is much more complex than the simple unimolecular case of synapsin. CaMKII often polymerizes into multimers of up to ten subunits, the activity of which is further affected by their respective phosphorylation states and presence or absence of calcium/calmodulin. It is through this mechanism by which action potentials are thought to be decoded and possibly amplified.

Matters of CaMKII modelling are further complicated because of the ability of CaMKII to trap calmodulin for a brief period even at sub-threshold calcium concentrations (Hanson et al ., 1994). This causes a frequency-dependent accumulation of holoenzyme activity dependent upon number of subunits and frequency of calcium spikes. The activity profile of CaMKII is therefore quite complex and unless a complete analysis of the priorities of the factors affecting CaMKII and its in vivo substrates is completed, the integration of such a model would prove useless as a model of an in vivo system. For the purposes of this project, therefore, the activity of CaMKII was reprented by the pair of static kinetic constants. In the case of low CaMKII activity, and therefore low levels of phosphorylated synapsin, a low k cat1 was selected relative to k cat2 , resulting in a lower population of phosphorylated synapsin versus non-phosphorylated synapsin. The result of this is evident in a visual comparison between figures 1 and 2. Note that in figure 2, the simulation of an increased population of phosphorylated synapsin yielded a significantly reduced population of cytoskeleton-bound synaptic vesicles. This supports the theory that a releasable pool of neurotransmitter may be maintained close to the presynaptic nerve terminal for regulated release by modulation of the phosphorylation state of synapsin present. In fact, previous studies involving gene knockouts in mice exhibited synaptic depression as a result of the lack of synapsin protein. This lack of synapsin protein precluded the maintenance of a releasable pool of neurotransmitter, and disallowed sustained exocytosis of neurotransmitter.

There are two possibilities speculated for the mechanism of neurotransmitter release involving the tethering of synaptic vesicles to the cytoskeleton by synapsin I. Actin filaments are often associated with mobility functions. This is evident from the polarity and nature of actin filaments and subunits. In fact, Bahler and Greengard (1987) described the ability of synapsin I to nucleate actin subunits and affect polymerization. It is therefore conceivable that synaptic vesicles are translocated within the cell via a “treadmill”-like mechanism involving polymerization and disintegration of filamentous actin within the cell. How the synaptic vesicles are able to translocate to the boundary of the nerve terminal when an action potential is to be decoded could result either by active transport of the synaptic vesicles tethered to the cytoskeleton, or by diffusion. Although diffusion is often a slow process over large distances, it is quite effective at microscopic scales and could be efficient enough for the translocation of synaptic vesicles. It is unlikely, however, that evolutionary pressure would allow to allow neurotransmitter to diffuse without strict directional control. Sudhof (1995) proposed, however, that translocation of synaptic vesicles was due to diffusion rather than active cytoskeletal transport due to the apparent dearth of cytoskeletal elements close to the active zones.

Conclusion

It is possible to construct a computer-based mathematical model of the dynamic interactions between synapsin phosphoproteins, synaptic vesicles, and the cytoskeleton. The model provides a simplified view of the binding kinetics involved. In order to determine whether or not the model accurately reflects a real-world system, it is possible to conduct further experimentation involving each of these components in vitro . Experimentation has been done using fluorescence resonance energy transfer to characterize the kinetics of synapsin-synaptic vesicle interactions, and it is possible to do the same with actin filaments. The computer model, however, provides a convenient method of projecting the effects of changes in phosphorylation state of synapsin based on changes in phosphorylation state. To further increase the relevance of this model, it is conceivable that a CaMKII model be integrated into the system, replacing the static kcat variables with a dynamic subroutine, which has its own activators and inhibitors.