Model of SNARE-Mediated Membrane Adhesion Kinetics

SNARE proteins are conserved components of the core fusion machinery driving diverse membrane adhesion and fusion processes in the cell. In many cases micron-sized membranes adhere over large areas before fusion. Reconstituted in vitro assays have helped isolate SNARE mechanisms in small membrane adhesion-fusion and are emerging as powerful tools to study large membrane systems by use of giant unilamellar vesicles (GUVs). Here we model SNARE-mediated adhesion kinetics in SNARE-reconstituted GUV-GUV or GUV-supported bilayer experiments. Adhesion involves many SNAREs whose complexation pulls apposing membranes into contact. The contact region is a tightly bound rapidly expanding patch whose growth velocity increases with SNARE density . We find three patch expansion regimes: slow, intermediate, fast. Typical experiments belong to the fast regime where depends on SNARE diffusivities and complexation binding constant. The model predicts growth velocities s. The patch may provide a close contact region where SNAREs can trigger fusion. Extending the model to a simple description of fusion, a broad distribution of fusion times is predicted. Increasing SNARE density accelerates fusion by boosting the patch growth velocity, thereby providing more complexes to participate in fusion. This quantifies the notion of SNAREs as dual adhesion-fusion agents.


Equations Governing SNARE Density Profiles
Let Γ s (r, t) and Γ c (r, t) denote the density of SNAREs and complexes, respectively, a distance r from the patch center. Note Γ c (r, t) = 0 for r > R p (t). The evolution equations read where the binding rate constant is only non-zero within the patch, Boundary conditions are: (i) zero SNARE complex flux across the moving patch boundary and (ii) the far-field density (at r = ∞ for large vesicles relative to patch size) is the initial SNARE density: After a transient a steady state patch growth velocity v patch results. For patch radius R p (t) much larger than the size of the region near the patch boundary where SNARE densities change substantially, the geometry becomes approximately 1D orthogonal to the patch boundary (x direction) and the far field boundary conditions are in effect at x = ±∞. In steady state the density fields are unchanging in a frame of reference moving with the boundary. This density field for the SNAREs is named Γ s (x) ≡ Γ s (r, t) where x ≡ r − v patch t and Γ c (x) for the complexes is defined similarly. Thus the general equations S1, S2, and S3 become the steady state equations 5, 6, and 7 in the main text, respectively. In eq. 7, the complex boundary condition deep inside the patch ("x = −∞") follows from SNARE number conservation.
Alternative Time Scale-Based Derivation of Scaling Results, Eqs. 8 and 11 In Exact Scaling Results of the main text a brief derivation of our scaling results for patch velocity in the slow and fast patch growth regimes was presented. More detailed derivations are given below, including derivation of the intermediate regime result. The arguments below are phrased in terms of key time scales, an alternative approach to the length-scale-based derivations in the main text.

Key time and length scales
The driving force for patch growth is the SNARE density in each vesicle, Γ snare . When this is far from the critical densities (eq. 11 of main text) the key timescales are well separated and v patch obeys exact power laws as a function of Γ snare . If the patch has a definite growth velocity v patch , there are 3 characteristic timescales: The first is the SNARE-SNARE binding time, the time required for all SNAREs at density Γ snare in one vesicle to undergo binding "reactions" in the patch region with cognate SNAREs in the apposing vesicle. τ s D is the SNARE diffusion time when SNARE diffusive displacement and patch boundary displacement just match ((Dτ s D ) 1/2 = v patch τ s D ); at longer times coherent patch motion beats diffusion. It can be thought of as the time a given SNARE initially near the interface remains close to it through its own diffusion before the interface moves onward. Similarly τ c D is the diffusion time of a SNARE-SNARE complex produced by a binding event. We will see the qualitative behavior depends on the relative magnitude of these scales.
Closely related to τ c D is the SNARE complex diffusion length, The outward osmotic pressure benefits only from complexes produced within l comp of the patch boundary.

Uncomplexed SNARE density profile in steady state
To self-consistently calculate the interfacial SNARE complex density setting velocity, the uncomplexed SNARE profile Γ s (x) must first be determined. Γ s (x) undergoes a transition from Γ snare far from the patch to zero deep inside the patch (Fig. 3). Suppose v patch is given (later, it will be determined selfconsistently from eq. 4 of the main text once Γ * c has been calculated). What is the SNARE density Γ * s at the patch boundary? Consider first a slowly growing patch, such that τ bind < τ s D : before the boundary has "moved on" there is sufficient time for all SNAREs inside the patch to undergo binding reactions. Thus a depletion hole in the SNARE profile Γ s (x) develops outside the boundary and Γ s is small inside the patch including the interfacial density, Γ * s Γ snare . The SNARE profile penetrates a small distance δ snare into the patch, equal to the distance SNAREs can diffuse before they bind after a time ∼ 1/k snare Γ * s , Note δ snare is determined by diffusion because on the binding timescale the boundary is effectively stationary since the diffusion time is much greater. The second relation above follows by number conservation: in steady state the rate SNAREs are consumed by binding reactions within the region of width ∼ δ snare equals the rate new SNAREs are input as the patch grows. Solving eq. S6 for the interfacial density gives The second result is in the opposite limit of a rapidly growing patch, τ bind > τ s D . In this case when a SNARE first enters the patch the probability it will bind before the boundary moves on is very small: the density is undepleted at the boundary, Γ * s ≈ Γ snare , there is no diffusion hole outside the patch and within the patch the SNARE profile tails off relatively gradually (see Fig. 3).
Slower patch growth, τ bind < τ s D Consider first SNARE densities low enough that the SNARE diffusion time exceeds the binding time. Then the SNARE interfacial density is small (see eq. 4, main text) as is the penetration depth of SNAREs into the patch, δ snare (Fig. 3b). The interfacial complex density is governed by the ratio of this scale to the complex diffusion length. Now from eqs. 4 and 11 of the main text and S5 and S6, this ratio obeys It follows that if the binding time is very small (τ bind < 2 τ c D ) or equivalently the SNARE density is less than a critical value Γ 1 crit , complex production is effectively a delta function source at the boundary, δ snare < l comp . All generated complexes can contact the boundary, Γ * c = Γ snare . Thus v patch = (kT /η d )Γ snare is the linear relation shown in eq. 8 of the main text.
At higher densities the complex diffusion length is less than the SNARE penetration depth. Of complexes generated in a region of width δ snare , a small fraction l comp /δ snare are within diffusive range of the interface and contribute to Γ * c : Using eqs. 4 (main text) and S4, this gives a different power law relation, v patch /v 0 = 2/5 µ 1/5 (Γ snare /Γ 0 ) 4/5 , as shown in eq. 8. This intermediate regime is valid up to the second critical density given by eq. 11, obtained by setting τ bind = τ s D and using the velocity law in eq. 8 (main text) for the intermediate regime.
Fast patch growth, τ bind > τ s D At high enough SNARE densities binding is relatively slow and the SNARE boundary density is undepleted, Γ * s ≈ Γ snare . The complex density at the boundary is the fraction of these SNAREs binding during the complex diffusion time After using eq. 4 of the main text and eq. S4, this gives v patch /v 0 = µ 1/3 (Γ snare /Γ 0 ) 2/3 , the high density regime of eq. 8.

Numerical Solution Methods
The steady state SNARE-mediated adhesion and patch growth kinetics eqs. 5, 6 and 7 of the main text were solved numerically as follows. First, a helpful result is derived. Assume v patch is given. Multiplying the differential equation for SNARE complexes (the 2nd of eq. 5) by e (v patch /Dc)x the equation is rewritten Integrating, one has Using the no-flux patch boundary condition in eq. 7 of the main text we obtain the following expression for the complex density at the patch boundary in terms of the uncomplexed SNARE profile Γ s (x): The power of this result is that the complex boundary density Γ * c can be obtained directly from the uncomplexed SNARE profile without the need to calculate the full complex profile Γ c (x). Our numerical procedure is as follows.
Step 1. A patch velocity v patch is assumed. The SNARE profile Γ s (x) is then calculated from the differential equation governing uncomplexed SNAREs, eq. 5 of main text, using a 4th-order Runge Kutta method.
Step 3. Γ * c is inserted into the drag law (eq. 4 of the main text for the linear drag law, or S14 for alternative law of next section) and solved for the patch velocity. We call this value v patch .
Step 4. If |v patch /v patch − 1| is sufficiently small, v patch is the self-consistent numerical solution. If not, a new improved value for v patch is selected and steps 1-4 repeated.

Results for Alternative, Non-linear Patch Drag Law
All results of the main text for adhesion and fusion kinetics were obtained assuming the drag force resisting patch growth depends linearly on the patch velocity v patch . This led to a simple linear local relation between velocity and patch boundary complex density, v patch ∼ Γ * c (eq. 4). Here we present results using the non-linear local relation proposed by de Gennes et al [32], which involves the vesicle tension γ and the viscosity η of the intermembrane fluid.
In effect, this relation results from a non-linear drag law Π drag = γ(v patch / v 0 ) 2/3 , where v 0 = γ/η, replacing eq. 3 of the main text. Balancing this with the SNARE complex osmotic pressure leads to the above result eq. S14. The authors of ref [32] suggest that dissipation due to fluid expulsion by the growing patch is dominated by fluid velocity gradients in a membrane wedge just outside the patch, and that increasing γ increases resistance by decreasing the wedge angle. This assumption may not be generally applicable but it is conceivable that the conditions required for this result's validity may be realized. We show below that using the alternative, non-linear local velocity-density relation of eq. S14 modifies the exponents in our predicted power law relations but otherwise leaves all conclusions qualitatively unchanged.

Adhesion Kinetics
The general forms of the key time and length scales are unchanged, and the same general expressions for the interfacial complex density Γ * c remain valid: Γ * c = Γ snare in the slow regime and the expressions of eqs. S9 and S10 in the intermediate and fast regimes, respectively. Substituting into these the new non-linear local velocity-density relation eq. S14 leads to new growth laws: Here Γ 0 and v 0 are characteristic density and velocity scales and the parameter λ is a dimensionless measure of SNARE binding reactivity, The three growth regimes are separated by critical SNARE density values The result for the slow regime was previously calculated in ref [32].
Figures S1 a and b show exact numerical solutions for patch growth velocity as a function of SNARE density in the slow/intermediate and fast regimes, respectively. Similarly to Fig. 5 for the linear relation, when velocity and density are appropriately scaled with critical values data for a range of and µ values collapses onto universal curves. The new power laws are confirmed by the numerical results.

Fusion Kinetics
The form of the fusion probability distribution is unchanged from the form of eq. 15 (main text) plotted in Fig. 6, and the expression of eq. 16 (main text) for the mean fusion time T fusion in terms of v patch and Γ snare remains valid. Using in this expression the new patch velocity-SNARE density results of eq. S15 gives where τ 0 = 0.88 (τ fus / v 2 0 Γ 0 ) 1/3 . These results apply provided steady state patch growth is reached and many SNAREs accumulate in the patch before fusion is triggered. These requirements are met if eq. 18 of the main text is satisfied. For low enough SNARE density eq. 20 of the main text is satisfied and fusion kinetics are described by the exponential distribution for single-SNARE fusion.