INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL RESULTS DISCUSSION REFERENCES

Fusion of enveloped viruses with either the plasma membrane or the endosomal membrane of host cells is mediated by specific viral transmembrane proteins. To this end, the ectodomain of these spike proteins has to undergo a conformational change into a fusion-active structure. Spike proteins of enveloped viruses taken up into a host cell by the endocytic route (as, for example, the hemagglutinin (HA) of influenza virus, the G-protein (VSV-G) of vesicular stomatitis virus, or the E1/E2-protein of Semliki Forest virus) undergo a conformational change at the acidic pH milieu in the late endosomal lumen. For spike proteins of viruses fusing with the plasma membrane of a host cell at neutral pH, e.g., the Sendai virus (F protein) or the human immunodeficiency virus (HIV) (gp41/120 protein), the trigger is thought to be set by interaction between the spike protein and receptors of the plasma membrane. Despite differences in the molecular architecture of the spike proteins, the natural target membranes, the uptake mechanism for the virus, and the trigger for the respective conformational change, it is generally believed that essential features of the fusion process are conserved among the various enveloped viruses. A general scheme of the fusion process has to envisage the following main steps. 1) The virus binds to the respective receptors of its target membrane. Often, and rather typically, binding and fusion activity are implemented in the same viral protein (e.g., HA, VSV-G, or gp41/120). 2) A fusion-active conformation of the viral fusion protein is triggered. Interaction with the receptor of the target cell may be essential for this conformational change and the conformation itself. Similar motifs of the secondary and higher structure of the ectodomain of different spike proteins have been identified under conditions where fusion is mediated. Several studies suggest that the structural feature of an extended, triple-stranded rod-shaped -helical coiled coil represents a common structural and functional motif of fusion proteins of various enveloped viruses (Skehel and Wiley, 1998) such as orthomyxoviruses (Carr and Kim, 1993; Bullough et al., 1994), paramyxovirus (Baker et al., 1999), retroviruses (Chan et al., 1997; Tan et al., 1997; Weissenhorn et al., 1997; Fass et al., 1996; Kobe et al., 1999; Caffrey et al., 1998), and filovirus (Weissenhorn et al., 1998; Malashkevich et al., 1999). The transition into a fusion-active structure is accompanied by the exposure of a hydrophobic fusion sequence inserting eventually into the target membrane (Durell et al., 1997). 3) A fusion pore is formed. Studies on influenza virus A (Morris et al., 1989; Ellens et al., 1990; Doms and Helenius, 1986; Danieli et al., 1996; Blumenthal et al., 1996) and baculovirus (Markovic et al., 1998) implicated that aggregation of the fusion proteins to a multimeric complex is required to form a fusion pore. Typically, to elucidate the intermediates of fusion pore genesis and their structures, fusion between viral-protein-expressing cells and appropriate target cells (e.g., red blood cells) is studied. Such a model system even allows detection of single fusion events between of two fusing cells.

Mathematical models may provide a powerful tool to simulate the fusion process, in particular between fusing cells, on a quantitative level and, by that, to understand common structural motifs and mechanisms of enveloped virus fusion (Bentz, 2000, 1992; Ludwig et al., 1995). Recently, employing a mass action kinetic model, the dynamics and size of the multimeric fusion HA complex was studied by comparing computed and measured fractions of cells equipped with the first fusion pore (Bentz, 2000). However, the predictive power of existing models is limited by the fact that a reasonable but nevertheless phenomenological function was used for the statistical distribution of the various time-dependent pore-forming events across an ensemble of fusing cells. Danieli et al. (1996) applied a Hill equation to model the dependence of the time delay in the onset of the lipid flow ( signal) upon varying HA densities. Blumenthal et al. (1996) assumed that the time shift between occurrence of a change of the membrane potential of the target membrane ( signal) and of the signal in single-cell fusion measurements follows an exponential probability distribution. Similarly, Bentz (2000) postulated a binomial distribution of first fusion pores across the population of cell-cell fusion complexes studied. Evidently, the fitted value of model parameters, e.g., the minimum number of aggregated HA trimers required to form a nascent fusion site, is strongly influenced by the choice of the statistical distribution function. Therefore, we propose here a model approach that aims at predicting the statistical distribution of the various time-dependent stages in pore formation. The approach explicitly takes into account the stochastic nature of the various elementary processes underlying the fusion process on the level of individual cell-cell contacts: the size of the contact area between two fusing cells, the lateral movement of fusion proteins and receptor molecules in the respective bilayer, and the interactions between fusion proteins and receptors.

Our approach does not allow only simulation of single cell-cell fusion events but also fusion measurements performed on cell suspension. The observed fusion kinetics measured in a cell suspension by monitoring time-dependent changes in the flow of fluorescent lipids results from the superposition of the stochastic cell-cell fusion signals. Accordingly, our modeling approach consists of two steps: 1) stochastic simulation of the fusion process taking place between single cells and 2) simulation of fusion kinetics in cell suspensions by using the ensemble average of individual stochastic trajectories.

The chain of events leading to the formation of a fusion pore between individual cells defines a Markov process that is governed by a master equation. The numerical solution of the master equation is performed by the algorithm introduced by Gillespie (1976) and provides stochastic time courses for the distinct intermediates of single-cell fusion. Values for the three unknown rate constants of the model are chosen such that a satisfactory concordance is achieved between simulated and observed and signals. Next, by averaging individual stochastic time courses across a sufficiently large number of cell-cell fusion complexes we demonstrate that the proposed model also provides a satisfactory simulation of the fusion kinetics measured in cell populations.

	MODEL

TOP ABSTRACT INTRODUCTION MODEL RESULTS DISCUSSION REFERENCES

Simulation lattice

The initial step of fusion pore formation consists in the binding of a HA-expressing cell to the target cell. Upon binding, small contact sites are formed between the cells, where both cell membranes lie approximately parallel and sufficiently close to each other to enable molecular interactions between them. These contact sites constitute the effective contact area. For what follows it is important to remark that the effective contact area can be considerably smaller than the total contact area estimated from micrographs of fusing cells. For example, Kozlov and Chernomordik (1998) reported the effective contact area between fusing cells to amount to ~1 µm². On the other hand, the total contact area has been assumed to be on the order of ~30 µm² (Frolov et al., 2000). In the following, we restrict the stochastic simulations to the kinetic processes taking place within the effective contact area. To this end, both membrane regions involved in a contact site are represented as a two-dimensional lattice constituted by small squared membrane units (in the following referred to as unit cells) covering the membrane area (Fig. 1). The edge length of a unit cell is set to 6 nm, which corresponds approximately to the spatial extension of HA trimers (see Wilson et al., 1981; Böttcher et al., 1999). Diffusion of HA trimers and receptors is modeled by random transitions between adjacent cells of the simulation lattice. The lattice is considered continuous, so that molecules leaving at one side are reintroduced at the opposite side, thus keeping the number of molecules constant. Lateral movement of integral membrane proteins is mainly brought about by random collisions with membrane lipids. The mean jump distance for membrane lipids is ~0.8 nm (Träuble and Sackmann, 1972). Thus, modeling lateral diffusion by random transitions between lattice cells with edge length of 6 nm fulfills the prerequisite that the mean free path of the diffusing molecule has to be smaller than the characteristic length of the simulation lattice.

View larger version (26K): [in this window] [in a new window]   FIGURE 1   Definition of the simulation lattice. (A) The effective contact area between fusing cells is considered to be formed by a number of contact sites representing a certain area within the interface between the cells where the two membranes are arranged in parallel to each other and in a sufficiently close distance allowing the formation of HA-receptor contacts. The arrows illustrate how the various macromolecular structures involved in the fusion process are mapped onto the simulation lattice. (B) Occupation state of the simulation lattice derived from the constellation in A. Generally, a unit cell of the simulation lattice can be occupied by one of the following variables: R, receptor molecule, freely diffusing; HA, HA (trimer) molecule, freely diffusing; HA*, HA (trimer) molecule belonging to a HA cluster (immobile); HA-R, unstable HA-receptor contact (immobile); C, tight HA-receptor-membrane link (immobile). Note that one HA trimer and one receptor molecule can simultaneously occupy one unit cell because they are actually located in two different membrane planes. Furthermore, note that the smallest lattice in simulation was of 50 × 50 unit cells. (C) Definition of an ion-permissive pore ( signal) and a lipid-permissive pore ( signal).

A single unit cell cannot be occupied by more than one molecule (HA or receptor R) in each of the respective membranes. The effective contact area is dissected into smaller simulation areas, each represented by a lattice of unit cells. As checked by successively increasing the size of the simulation lattice, 50 × 50 unit cells are enough to prevent a notable influence of marginal effects caused by re-entering fusion intermediates leaving the simulation lattice during the simulation. Although the simulation lattice was initially chosen for computational reasons, experimental data underline that this size is in good agreement with that of a single contact site.

The cumulative distribution function F_tot(t) for any fusion intermediate to occur within the time span t in the effective contact area between two fusing cells can be related to the corresponding distribution function F_sample(t) for a simulation lattice by

(1)

where N refers to the number of simulation lattices required to cover the effective contact area. Equation 1 can be easily rationalized by considering that p = 1  F_sample(t) is the probability that a fusion intermediate has not occurred within time span t in a single simulation area. Hence, the probability that this intermediate has not occurred within this time span in any of the N independent simulation areas is given by p^N.

Following Kozlov and Chernomordik (1998), the total contact area between fusing cells amounts to ~1 µm². Based on this value, the distribution function (Eq. 1) has to be calculated from the distribution function of N = 12 (12 × 0.09 µm²  1 µm²) simulation lattices.

Model variables and processes

The variables of the fusion model and the elementary processes that may take place between them are detailed in the following.

HA: activated hemagglutinin trimers

R: receptor

HA-R: HA-receptor contact

HA*: immobilized HA trimer captured in a HA cluster

C: HA-receptor-membrane link

IP: ion-permissive early fusion pore

LP: lipid-permissive pore

Master equation

Let p denote the probability to find a molecule of type X (X = HA, R, HA*, HA-R, or C) in the unit cell (i, j) at time t (the time argument will be dropped in the following). The subscripts i and j (i, j = 1, ... , 50) label the row and column of the simulation lattice. Generally, the time evolution of the probability is governed by the master equation:

(2)

The linear time-evolution operator A(i, j, Xi', j', X') in Eq. 2 gives the probability with which occupation of the unit cell (i, j) by the molecular species X is affected by the molecular species X' resident in the unit cell (i', j'). Because the kinetic processes included in the model may take place either in a single cell (formation of HA-R or C) or between adjacent cells (lateral diffusion of HA, formation of HA* clusters) the summation across the cell indices i' and j' in the master equation (Eq. 2) actually covers only i' = i, i + 1, i  1 and j' = j, j + 1, j  1. For example, occupation of the unit cell (i, j) by a mobile HA trimer may increase from 0 to 1 only due to invasion of a HA trimer from one of the neighboring unit cells or dissociation of a HA-receptor contact present in the same cell. Correspondingly, occupation of the unit cell (i, j) by a mobile HA trimer may decrease from 1 to 0 by transition of a HA trimer from this cell into one of the neighboring unit cells or by formation of HA-receptor contact within this cell (for rate constants refer to Table 1):

(3.1)

Similar equations hold for the time-dependent change of the probabilities for the other variables of the model:

(3.2)

(3.3)

(3.4)

Simulation technique

Numerical solution of the master equation (Eq. 2) can be carried out by means of a compact and simple simulation algorithm (Gillespie, 1976).

1) Generating random initial occupations

2) Choosing a random time step

Atott

3) Selecting randomly a distinct transition process

4) Updating the occupation numbers of the unit cells

Stochastic simulations of single-cell fusion kinetics

The master-equation approach outlined above provides values for the stochastic variables t and t representing the delay times for the first occurrence of an ion-permissive fusion pore and of a lipid-permissive pore in the simulation lattice. Repeating the stochastic simulation for a sufficiently large number (N_sample) of independent simulation lattices one may generate sets of stochastic delay times, {t} and {t}, n = 1, ... , N_sample, to derive cumulative frequency distributions:

(3)

(4)

whereby (x) denotes the unit-step function; i.e., (x) = 1 if x  0 (0 else). By employing Eq. 1, these cumulative frequency distributions for the first occurrence of a fusion-pore intermediate (ion-permissive pore or lipid-permissive pore) on the simulation lattice can be used to calculate the related cumulated frequency distributions F(t) and F(t) for the total effective contact area.

Stochastic simulations of fusion kinetics in cell suspensions

The cumulative distributions F(t) and F(t) count the frequency (or probability) for a fusion intermediate to occur in a single-cell fusion experiment within the time span t after initiation of the fusion process. These distributions can be used to provide large sets of stochastic delay times, t and t, for the occurrence of an early fusion pore or a lipid-permissive pore in the -th cell-cell contact. Given that each lipid-permissive pore initiates lipid mixing, the overall dequenching signal (FDQ) observed in a population of cells (N_cells) at time t represents the superposition of dequenching signals initiated at time t by the -th cell pair:

(5)

The time course for fluorescence dequenching due to formation of a lipid-permissive pore between a single pair of fusing cells at time t₀ can be roughly described by the function

(6)

where  characterizes the kinetics of the lipid redistribution (Chen et al., 1993). Hence, the overall dequenching signal can be written as

(7)

	RESULTS

TOP ABSTRACT INTRODUCTION MODEL RESULTS DISCUSSION REFERENCES

Simulation of single-cell fusion kinetics

The proposed mathematical model was first applied to simulate single-cell fusion kinetics as measured by Blumenthal et al. (1996). The numerical values of the kinetic parameters used in these simulations are depicted in Table 1. The rate constants for the diffusion of HA trimers and receptor molecules were calculated from measured diffusion coefficients D (Danieli et al., 1996; Gutman et al., 1993) according to D = x²/4t, yielding k = 4D/ x² with x = 6 nm for the rate with which a transition takes place from one unit cell to the neighboring unit cell. Numerical values for the remaining three rate constants were chosen such that a reasonable concordance between simulated and observed data was achieved. Note that the definition of the rate constants used in the model is different from the definition of phenomenological rate constants in chemical reaction kinetics in that the latter also include the collision probability (cross section).

Fig. 2, A and B, depict the distribution of HA trimers across the simulation lattice at the beginning (t = 0 s) and after 50 s of simulation. Placing the HA trimers randomly to the cells of the simulation lattice at t = 0 s, most of them are isolated without forming clusters with adjacent HA trimers. During the fusion process almost all HA trimers are trapped into immobile HA-clusters of different shape. It has to be noted that irrespective of this clustering, the HA trimers keep interacting with receptor molecules to form HA-R contacts and HA-receptor-membrane links (not shown in Fig. 2, A and B).

View larger version (22K): [in this window] [in a new window]   FIGURE 2   Simulation results for a 50 × 50 simulation lattice. (A) Example of the initial distribution of HA trimers across the simulation lattice. (B) Distribution of HA trimers across this simulation lattice after t = 50 s. Most of the HA molecules are trapped into immobile HA clusters. HA aggregation does not affect interactions with receptor molecules; i.e., HA trimers trapped into a cluster may still form weak HA-R contacts and stable HA-receptor-membrane links. (C) Example of stochastic trajectories for free receptors (R), HA-R contacts, and stable HA-receptor-membrane links (C). The vertical lines indicate the delay time tIP = 134 s for the first occurrence of an ion-permissive pore ( signal) and tLP = 605 s for the first occurrence of a lipid-permissive pore ( signal) during the simulation.

Fig. 2 C shows an example of the time-dependent stochastic trajectories for free receptors, HA-R contacts, and stable HA-receptor-membrane links for one simulation on a 50 × 50 lattice. There is a decline in the number of free receptors from initially 65 molecules to ~20 molecules after 600 s. This decline is paralleled by a increase in the number of stable HA-receptor-membrane links hosting most of the available receptors because the number of HA-R contacts remains very small during the whole time course. The reversibility in the formation and decay of HA-R contacts is reflected by large fluctuations of the respective trajectory. The vertical lines in Fig. 2 C indicate the delay time t_ip = 134 s for the first occurrence of an ion-permissive pore ( signal) and t_lp = 605 s for the first occurrence of a lipid-permissive pore ( signal) during the simulation.

From repeated simulations on a 50 × 50 simulation lattice we constructed the statistical distributions (Eqs. 3 and 4) of the characteristic time span needed for the first occurrence of an ion-permissive pore and of a lipid-permissive pore in a simulation lattice. Next, the respective distributions for the total contact area were computed by Eq. 1 using N = 12, i.e., considering the contact area to be constituted by 12 contact sites. The obtained theoretical distributions of signals (first occurrence of an ion-permissive pore) and signals (first occurrence of a lipid-permissive pore) are both in reasonable concordance with the measurements (Fig. 3). It is seen, however, that the calculated cumulative distribution of signals exhibits a slightly steeper ascent than indicated by the measurements. This slight but systematic discrepancy could be possibly attributed to the choice of the number of contact sites, N, and the minimum number of HA trimers involved in the pore formation (see Discussion).

View larger version (21K): [in this window] [in a new window]   FIGURE 3   Kinetics of single-cell fusion: cumulative distribution of delay times for signals (first occurrence of an ion-permissive pore) and signals (first occurrence of a lipid-permissive pore). Closed symbols refer to the experimental data of Blumenthal et al. (1996). Open symbols denote the simulated distributions based on the kinetic parameters in Table 1. The theoretical distributions have been obtained for a relative HA and receptor density of 3% occupation of the simulation lattice corresponding to 750 molecules per µm2 of membrane area. The construction of the cumulative distributions is based on a total of 674 simulation trials on the simulation lattice. Only those signals have been taken into account that took place within a total simulation interval of 3000 s on a lattice.

Variation of hemagglutinin and receptor densities

In a second series of simulations we studied the effect of varying concentrations of HA trimers and receptor molecules on the distribution of the delay times t_IP and t_LP for the first occurrence of an ion-permissive pore and a lipid-permissive pore (Fig. 4, A-D)_. The average occupation of the simulation lattice by HA trimers and receptor molecules was varied between 1% and 12%, which corresponds to actual molecule densities per membrane area of 250-3000 molecules/µm². As expected, the time delay in the formation of fusion intermediates becomes shorter with increasing concentrations of HA trimers and receptor molecules. According to our simulations, increasing the receptor density should be slightly more efficient in accelerating the fusion process than a comparable increase in HA density.

View larger version (35K): [in this window] [in a new window]   FIGURE 4   Kinetics of single-cell fusion: influence of varying HA densities (A and B) and receptor densities (C and D) on the cumulative distribution of delay times for signals (first occurrence of an ion-permissive pore (A and C)) and signals (first occurrence of a lipid-permissive pore (B and D)). The cumulative distributions have been calculated at relative HA and receptor densities of 1%, 3%, 9%, and 12% occupation of the simulation lattice corresponding to 250, 750, 2250, and 3000 molecules per µm2 of membrane area. More than 600 simulations contribute to each curve.

Simulation of fusion signals observed in cell suspensions

A standard technique to trace the time-dependent formation of lipid-permissive pores ( signal) in cell suspensions consists in monitoring the fluorescence dequenching associated with the inter-membrane redistribution of a lipid dye. As outlined above (cf. Eq. 7) the dequenching signal at time t represents a superposition of signals initiated at different times determined by stochastically distributed cell-cell fusion events. Fig. 5 shows a typical time course of monitored during fusion between HA-expressing fibroblasts and red blood cells (see Fig. 3 of Danieli et al., 1996). The simulated time course in Fig. 5 was obtained according to Eq. 7 by applying the cumulative distribution of delay times t_LP for the first occurrence of a lipid-permissive pore obtained in the simulations of single-cell fusion events outlined above. The best fit to the experimental data was obtained by putting the rate constant for the kinetics of the lipid redistribution to the value  = 350 s. The excellent concordance between the simulated and measured time course of demonstrates that the results obtained in the simulations of single-cell fusion events are fully consistent with kinetic data obtained in cell suspensions.

View larger version (19K): [in this window] [in a new window]   FIGURE 5   Kinetics of lipid mixing ( signal) in suspensions of fusing cells. The open squares refer to the fluorescence dequenching in suspensions of HA-expressing cells and red blood cells at 28-29°C measured by Danieli et al. (1996; see Fig. 3 of this paper). The solid line was obtained on the basis of Eq. 7 using the simulated distribution of delay times tLP for the first occurrence of a lipid-permissive pore shown in Fig. 3. The rate constant for the kinetics of lipid redistribution was set to  = 350 s.

Based on the results shown in Fig. 4, we also simulated the influence of varying HA and receptor densities on the dequenching () signal in cell suspensions (Fig. 6). The average occupation of the simulation lattice by HA trimers and receptor molecules was chosen to 1%, 3%, 9%, and 12% corresponding to actual molecule densities per membrane area between 250 and 3000 molecules/µm². As a measure for the lag period seen between initiation of fusion pore formation by acidification and notable increase of FDQ, we determined from the simulated time courses the time t* at which 20% of the FDQ signal had occurred. Plotting the inverse lag time as a function of HA or receptor density indicates saturation with increasing densities (Fig. 6, C and D). The relative shortening of the lag period achieved by increasing the density of HA trimers from 1% to 12% is ~1.8. This value is of the same order as reported by Danieli et al. (1996) who have varied the HA content in the membrane by using different HA-expressing cell lines. Remarkably, our simulations predict a more significant decrease in the lag time (about three-fold) when increasing the density of receptors in the same range (1% to 12%) as the HA trimers.

View larger version (27K): [in this window] [in a new window]   FIGURE 6   Influence of varying HA and receptor densities on the kinetics of lipid mixing ( signal) in suspensions of fusing cells. (A and B) The time courses have been calculated on the basis of Eq. 7 using the simulated distribution of delay times tLP obtained at relative HA and receptor densities of 1%, 3%, 9%, and 12% occupation of the simulation lattice corresponding to 250, 750, 2250, and 3000 molecules per µm2 of membrane area. The lag time t* is defined as the time at which 20% of the FDQ signal has occurred. (C and D) Plot of the inverse lag time (1/t*) at four different densities of HA trimers and receptors.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MODEL RESULTS DISCUSSION REFERENCES

To simulate the HA-mediated formation of a fusion pore we have developed a mathematical model that explicitly takes into account the stochastic nature of the molecular events underlying the fusion process. The proposed model encompasses the following steps reported in the literature to be included in fusion pore kinetics: 1) lateral diffusion and self-aggregation of the fusogenic viral membrane protein oligomers (trimers), 2) lateral diffusion of the receptor molecule in the target membrane, 3) formation of noncovalent contacts between HA trimers and receptors, 4) irreversible conversion of these contacts into tight links between HA and the target membrane, and 5) clustering of HA-receptor contacts and HA-receptor-membrane links leading to the formation of early states of the fusion pore characterized by ion conductivity and a lipid flow between fusing cells, respectively.

A particular advantage of our approach is the possibility to model fusion signals measured in cell suspensions as a superposition of stochastic fusion signals arising from single-cell fusion events. This is the first successful attempt to model consistently both types of fusion kinetics. The fact that both types of experiments could be reasonably well described by choosing appropriate values for only three adjustable kinetic parameters (k₊, k, and k_C) underlines the reliability of the proposed model.

Validating the model

The contact site was assumed to be a lattice of small squared membrane units (6 × 6 nm) corresponding to a size of 300 × 300 nm. This is of the same order as the contact size between HA-containing liposomes deduced from quick-freezing electron microscopy (see Fig. 11f in Kanaseki et al., 1997). Furthermore, from electron microscopy images of Frolov et al. (2000) it can be roughly estimated that the radius of the membrane surface area involved in the formation of direct contacts between erythrocytes and HA-expressing cells is on the order of 100-150 nm. Due to the larger size of attachment area between fusing cells (Danieli et al., 1996; Frolov et al., 2000) the existence of several contact sites is a reasonable assumption.

Numerical values for the kinetic parameters of the model have been either taken from the literature or estimated by comparing simulation results with experimental data. A value of ~3 mM was estimated for the dissociation constant of the sialic acid-HA ectodomain complex (Sauter et al., 1992). Based on this value it was concluded that on the average the percentage of HA trimers and receptors temporarily involved in HA-R contacts should be lying in the range 60% to 98% (Leikina et al., 2000). This estimate is in good agreement with our theoretical finding that for a lattice occupation of 3% HA trimers and 3% receptor molecules (corresponding to a normal membrane density of 750 molecules/µm²) the percentage of HA trimers and receptors involved in HA-R contacts was ~53% during the initial phase of the stochastic simulations.

The existence of an early fusion pore was evidenced by cell membrane capacitance measurements and by lipid dyes sensitive to the membrane potential (Blumenthal et al., 1996; Zimmerberg et al., 1994; Tse et al., 1993). In the model, an early fusion pore is defined as a cluster of three HA-R contacts in triangle configuration. Based on this assumption the model provides a satisfactory statistical distribution of single-cell signals. Nevertheless, it cannot be excluded that the number of HA-R contacts constituting an early fusion pore is different from three (see below).

In the model, formation of HA-R contacts is described as a reversible process. Hence, ion-permissive pores may decay again. In the simulations this is reflected by random fluctuations in the number of early fusion pores. Such instability of the ion-permissive pore was indeed elucidated by a flickering of its conductance (Melikyan et al., 1993; Spruce et al., 1991; Zimmerberg et al., 1994). A further important consequence implied by the possible decay of HA-R contacts is that the first lipid-permissive pore (defined as a cluster of stable HA-R cross-links) does not necessarily emerge from the first early fusion pore.

Experimental studies have shown that the essential conformational change of HA into a fusion-competent state is supported by HA-receptor interactions (de Lima et al., 1995; Stegmann et al., 1995; Leikina et al., 2000). To describe adequately this observation, in the model, formation of a stable HA-receptor-membrane link proceeds obligatorily via a reversible HA-R contact; i.e., there is no direct transition HA + R  C. The nature of this essential conformation change of activated HA has still to be identified. Following recent studies one may suggest that this transformation reflects later stages of the refolding of HA. For example, this transition might be accompanied by the formation of the extended coiled-coil motif and/or of an anti-parallel helices bundle (Bullough et al., 1994; Chen et al., 1999) facilitating or mediating the close approach of membranes necessary to form a stable fusion pore. The estimated half-time value for the transition rate HA-R  C amounts to ~5 s (ln2/k_p). In our model, the transition rate refers to the conformational change. Of course, the time required for formation of the first ion-permissive pore is much longer. Bentz (2000) found a rate of 10⁴ s in his kinetic fusion model as the characteristic time required for the conformational change to occur after triggering of the fusion process. This value includes all processes preceding the conformational change, for example, aggregation of HA trimers and formation of HA-receptor contacts, which in our model are considered separately.

The lipid-permissive pore is commonly regarded as an important intermediate of fusion pore formation. In the model, the lipid-permissive pore requires three HA trimers arranged in triangle geometry. This assumption was essentially based on experimental observations of Danieli et al. (1996) who concluded that membrane fusion requires the concerted action of at least three activated HA trimers. In contrast, Blumenthal et al. (1996) interpreted their experiments on the basis of a theoretical model that yielded about six HA trimers to be necessary for fusion. It has to be considered that the latter estimate was derived from kinetic data on the formation of solute-permissive fusion pores, whereas Danieli et al. (1996) derived their estimate from measurements of lipid mixing. We have not studied in detail how changes in the assumption on the minimum configuration of an ion-permissive pore and a lipid-permissive pore may affect the quality of the simulations with respect to available experimental data.

Aggregation of HA trimers to larger clusters is thought to be caused by interactions of hydrophobic sequences of the HA trimer. It has been concluded that apart from the N-terminus of the HA ectodomain other hydrophobic sequences become also exposed after activation of HA by the low-pH trigger (Korte and Herrmann, 1994; Burger et al., 1991). A rapid and irreversible aggregation of HA on cells is consistent with experimental data (Ellens et al., 1990; Melikyan et al., 1995a; Danieli et al., 1996; Gutman et al., 1993). From photobleaching experiments, it was concluded that aggregation of HA trimers results in a suppression of fusion (Gutman et al., 1993). In the model, HA aggregates are treated as completely immobilized but without becoming inactivated, i.e., without losing their competence to interact with receptors and to form stable HA-receptor contacts. This is a reasonable assumption, keeping in mind that the experimental data that served as the basis for comparison with the model are based on HA-expressing cells from the influenza virus A Japan strain (Danieli et al., 1996; Blumenthal et al., 1996). It has been shown that inactivation of HA from the Japan strain of influenza virus (A/Japan/305/57) is a very slow process (Puri et al., 1990; Korte et al., 1999). Moreover, neglecting HA inactivation as a kinetic relevant process in a contact site seems to be justified by the observation that HA inactivation is suppressed in the presence of the target membrane (Ramalhosantos et al., 1993; Leikina et al., 2000).

Simulation of single-cell fusion kinetics

In our simulations of single-cell fusion kinetics we could achieve a good overall concordance with the experimental data of Blumenthal et al. (1996). Nevertheless, the computed cumulative distributions for the first occurrence of signals appear to be slightly steeper than the corresponding experimental distributions. This discrepancy can be possibly accounted for by the fact that we used a fixed number of N = 12 simulation lattices of size to represent the effective contact area (1 µm²) between fusing cells. Actually, the size of the contact area itself is a stochastic variable so that the number N of simulation lattices and hence the shape of the cumulative distribution calculated on the basis of Eq. 1 will vary from one single-cell complex to the other. Thus, the relative contact area, i.e., the membrane fraction actually involved in the formation of inter-membrane contacts, turns out to be an essential parameter of the model.

The rate of fusion pore formation depends on HA and receptor density

The model allows us to address the influence of distinct parameters on the fusion kinetics. For example, from experiments, opposite conclusions were drawn on the role of the HA-receptor interactions in fusion. Alford et al. (1994) observed a decline of influenza virus fusion at high concentrations of sialic-acid-bearing gangliosides in the liposomal target. They concluded that only those HA trimers can form a fusion pore that are not associated with receptors of the target membrane (see also Ellens et al., 1990). Here, simulating the effect of increasing HA densities (at constant receptor density) we found a significant increase in the overall rate of the fusion process. This result is in agreement with experimental findings (Danieli et al., 1996; Clague et al., 1991). Similarly, increasing the receptor density (at constant HA density), the model predicts an acceleration of the fusion process. Several independent studies arrived at the conclusion that HA-receptor complexes are important for cell fusion (see above).

Nevertheless, it is known that HA can trigger fusion of influenza viruses with lipid membranes bearing no receptor (Schoen et al., 1996). Thus, in principal, one has to consider that also HAs not associated with a receptor can be involved in triggering membrane fusion. Such a mechanism could be implemented in our approach. However, we did not include this mechanism for several reasons. First, our study is to our knowledge the first approach describing protein-mediated fusion as a stochastic process. To provide a clearly arranged model, we kept the number of processes/steps low. Second, a goal of our work was to develop a model applicable also to fusion of other viruses. Notably, for fusion of other enveloped viruses, e.g., human immunodeficiency virus, interaction with receptors is essential for fusion-mediating viral proteins to transform into their fusion-active conformation. Third, we consider the rather high part of HAs bound to receptors (see above) as a justification to neglect a fusion mediated by HAs not bound to receptors.

Simulation of fusion experiments with cell populations

Based on our stochastic approach we have simulated the kinetics of lipid mixing between fusing HA-expressing cells and red blood cells in suspension. The time course of lipid mixing can be represented as a convolution of two distinct kinetic processes (Chen et al., 1993): 1) initiation of lipid mixing due to the formation of a lipid-permissive pore, triggering 2) redistribution of the fluorescent lipid-like probe between the fusing membranes. The latter process can be monitored by fluorescence dequenching. Chen et al. (1993) have shown that lipid (probe) redistribution between two fusing cells joined by a narrow neck can be approximated by a single exponential (see Eq. 6) decaying with a characteristic time . Choosing a value  = 350 s for the rate constant of lipid redistribution we obtained a surprisingly good concordance of these simulations with (cuvette) experiments carried out at 28-29°C (Danieli et al., 1996). Chen et al. (1993) have developed a theoretical approach to relate the parameter  to the size (radius) of the two cells, the lateral diffusion coefficient of the probe, and the radius of the fusion pore. The size of the early fusion pore should be in the range 10-15 nm because the thickness of each bilayer is ~5 nm, and the inner diameter of the early fusion pore is ~4 nm (Kanaseki et al., 1997). The radius of the HA-expressing fibroblasts is ~11 µm (see Fig. 1 A in Danieli et al., 1996), and the radius of red blood cells is ~3 µm. Taking these values and a lateral diffusion coefficient of 0.5 × 10⁸ µm²/s for R18 (extrapolated from the value of 0.3 × 10⁸ at 22°C (Aroeti and Henis, 1986), the formalism of Chen et al. (1993) yields  = 420 s, which is very close to the fitted value of  = 350 s.

In summary, our model provides a satisfactory quantitative simulation of kinetic data on HA-mediated fusion pore formation gathered by different types of kinetic measurements. Further progress in mathematical modeling will depend upon the availability of reliable kinetic data (rate constants) for the various elementary processes. If so, other steps and intermediates of the HA fusion process as, for example, the well established hemifusion intermediate (Kemble et al., 1993; Melikyan et al., 1995b; Nussler et al., 1997) can be integrated into simulation. Nevertheless, at the current state of experimental research, the proposed model appears to be consistent with most kinetic features of fusion pore formation documented in the literature. Finally it has to be pointed out that the proposed stochastic approach is also applicable to other types of membrane-membrane interactions such as, for example, cell-cell attachment.