Our model captures coarse structure of the Gag and Gag-Pol monomers as derived from a recent cryoET structure (Schur et al., 2016) of the immature lattice (Figure 1A) (Methods). The Gag-Pol is structurally identical to the Gag but represents 5% of the total monomer population to track protease locations within the lattice. We were able to assemble a variety of spherical Gag lattices that grew from monomers to a single sphere with our targeted coverage of the membrane surface using our stochastic reaction-diffusion simulations (Varga et al., 2020; Figure 2) (Methods). The lattices in Figure 2 are a single connected continent, with imperfect edges, a large gap on the surface (~1/3), and regions with defects present in the tri-hexagonal lattice (Figure 2). Our lattice topologies are in very good agreement with the structures determined by cryoET, which also shows a single continent and a large gap, such that the spherical lattice is truncated (Tan et al., 2021). We quantified the fraction of hexamers in our lattices that are incomplete, finding 36–40% have fewer than 6 monomers when binding events during assembly are irreversible, or 30–32% when we allow unbinding during assembly (see Methods). This is in excellent agreement with the 34%±4% we calculated from the cryoET datasets (Tan et al., 2021). We also observe a similar distribution in the sizes of the regions containing incomplete hexamers, with most regions being localized and small, but with a few larger strands or ‘scars’ (Figure 2—figure supplement 1). Along the incomplete edge, we count a larger fraction of the free binding sites are hexamer sites, in agreement with experiment (Tan et al., 2021), although we acknowledge our simulations do not exclude free dimer sites (which are not observed in the cryoET) given the assembly parameters (dimer and hexamer rates are equally fast).

It is illuminating that the structures of our lattices share features with the experimental lattices, given that our assembly simulations (see Methods) do not directly mimic the physiologic process of Gag assembling in the cytoplasm, at the plasma membrane, with RNA (Tritel and Resh, 2000). To promote the nucleation and growth of only a single lattice (rather than nucleating multiple lattice structures), we combined fast Gag-Gag binding (6×10⁶ M^–1s^–1) with a slow titration of Gag monomers into the volume. The slow titration does mimic the role of co-factors, however, in that Gag does not assemble without being effectively ‘turned on’ by co-factors like RNA (Qian et al., 2023). The similarity of our structures to experiment suggests that our assembled model is constrained to incorporate topological defects at a similar frequency to the biological proteins. Interestingly, while these lattices must have defects because a sphere cannot be perfectly tiled by a hexagonal lattice, the number of non-hexamers or imperfect contacts within them is significantly higher than the number required by Euler’s theorem, which is only 6 for a spherical lattice with a hole in it (Negri et al., 2015). We speculate that during assembly, the lattice is not undergoing a significant amount of remodeling and annealing to correct these defects. This would be consistent with a fast and more irreversible nucleation and growth, and indeed we see fewer defects (~31% vs 38%) when we allow for unbinding during assembly vs irreversible binding. The biological lattices seem to be ‘good enough’ despite the possibility of more perfect lattice arrangements, and the lower stability of these more defective lattices should facilitate the remodeling necessary for maturation.

Although only 5% of the Gag monomers in our simulation are tagged as Gag-Pol (~125 out of ~2625 simulated proteins), we find it is extremely unlikely that a lattice will be assembled without a pair of them already adjacent (Figure 2B). This is due to the stochastic nature of the assembly and the fact that each monomer has 3 adjacent monomers, two via its hexamer interfaces and one via its dimer interface. However, we can reduce the number of Gag-Pol to Gag-Pol pairs if we turn off any specific interaction between them by setting their binding rates to 0. Even making this interaction thus highly unfavorable relative to a Gag to Gag or Gag to Gag-Pol interaction, we still find pairs of them adjacent, as they can be brought into proximity via their specific interactions with the Gag monomers (Figure 2B). The number of pairs given the unfavorable interaction is also dependent on the assembly conditions; when we allow for unbinding between Gag contacts this allows for annealing and correction of such unfavorable contacts during assembly, and the Gag-Pol to Gag-Pol pairs are largely eliminated. Overall, these results indicate that to prevent early activation of the proteases, one cannot just rely on the lower frequency of Gag-Pol to Gag-Pol interaction, as the lattice is simply too densely packed. Instead, these Gag-Pol dimers would have to be actively inhibited from initiating protease activity by either having a highly unfavorable affinity for one another or otherwise forming dimers that are enzymatically inhibited, as any activation preceding budding can leak proteases back to the cytoplasm (Bendjennat and Saffarian, 2016), and is known to reduce infectivity (Kräusslich, 1991). Regardless of how the activation is prevented, inhibition would have to be released following budding, and this mechanism is not known. We assume below that the Gag-Pol to Gag-Pol dimers following budding can now interact favorably, identically as Gag to Gag, since we know that activation must ultimately occur.

We perform all our simulations from the same starting structures, but with a range of hexamer strengths of –5.62 to –11.62kBT, and a slower (0.015 μM^–1s^–1), medium (0.15), and faster (1.5) rate of binding for each ΔGhex . For the weakest hexamer contacts of –5.62kBT, we find that the lattice is not able to retain its single continental structure, and instead fragments into a distribution of much smaller lattices (Video 2). Given a fixed ΔGhex we speed up the on- and off-rates and as expected, we see more rapid disintegration of the lattice structure. As we stabilize the lattice by increasing ΔGhex , we still see departure from the single continental structure due to unbinding of monomers and small complexes from the lattice edge (Figure 3). Hence, we see the emergence of a bimodal distribution of lattices, with a peak at the monomer/small oligomer end, and another peak containing the majority of the lattice in one large continent. The size of the large continent remains largest with increasing ΔGhex and with a slower rate, over the course of these ~17–20 s simulations (Figure 3). Importantly, these dynamics occur in all our simulations and would not be possible if not for the incompleteness of the lattice. Specifically, the Gag contacts are dissociating not from the membrane but from each other, predominantly along the edge, at which point they can then diffuse along the membrane surface (Video 1, Video 2, Video 3). If the lattice were covering 100% of the surface, dissociation events would not allow Gags to diffuse away, and no dynamic remodeling would occur. From the sizes of the lattices present in the simulations, we can also report on the distribution of diffusion constants represented on the surface, as larger lattices diffuse more slowly. For the weaker lattices, the distribution is very broad, spanning 4 orders of magnitude, whereas for the most stable lattice there is primarily one very slowly diffusing timescale, and a separate timescale for the more faster moving oligomers (Figure 3—figure supplement 1). Lastly, the lifetimes of hexamers in our lattices are controlled by ΔGhex , by a ΔGstrain penalty, and by the extent to which the hexamers are constrained by further dimer contacts. Our ΔGstrain penalty is small at 2.3kBT but it does shorten the hexamer lifetimes relative to having 0 strain (Methods). This means that the strain penalty can increase lattice dynamics, but we see that the relaxation dynamics from the initial lattices is much more sensitive to the magnitude of ΔGhex (Figure 3). The effect becomes negligible for more stable lattices as the remodeling we observe is dominated by Gag subunits on the edge that form incomplete hexamers. Overall, if the value of ΔGstrain were large (~ΔGhex) it could impact at which value of ΔGhex the lattice transitions from a primarily single-connected component to the fragmented lattice we see here at ΔGhex=–5.62kBT.

A primary goal is to characterize the path and the timescale by which two Gag-Pol molecules could find one another given the single continental lattice they are embedded in at time 0. From our initial lattices, we showed in Figure 2 that at least one pair of Gag-Pol monomers are already in contact with one another, so we ignore those pairs to focus instead on spatially separated Gag-Pols. By tracking the separation between all pairs of Gag-Pol monomers (Figure 4), we can quantify the first-passage time (FPT), or the time for the first pair to find one another in each simulated stochastic trajectory. The edge of the incomplete lattice supports multiple detachment events (Video 1); of the ~250 Gag monomers along the edge, 10% have only one link to the lattice, which offers the easiest path to disconnect, by breaking only a single bond (Figure 4—figure supplement 1).

In Figure 4, our results show how the FPT for two Gag-Pols to dimerize with one another is dependent on both unbinding rates and binding rates, as both events are required to bring two Gag-Pol together. We observe two intuitive trends. One is that for a given free energy ΔGhex, a faster association (and thus faster dissociation) rate results in faster dimerization events between the Gag-Pol monomers. The second trend is that as the lattice free energy stabilizes, dimerization events are slowed due to the slower dissociation times, despite having the same on-rates (Video 3). These timescales are thus consistent with the dimerization events requiring at least one of the monomers to dissociate from the lattice, and then rebind at a new location containing a Gag-Pol. We report the association rates as their 2D values, because binding is occurring while the proteins are affixed to the 2D membrane surface, as unbinding from the surface is rare (Methods). The corresponding 3D rates are representative of slow to moderately fast rates of protein-protein association (1.5×10⁴–1.5×10⁶ M^–1s^–1), where they are converted to 2D values via a molecular length-scale h=10 nm (Methods). Activation requires explicitly that the 5% of monomers carrying the proteases to be involved. Hence, additional unbinding and rebinding events will occur that are not ‘activating’ because they involve a Gag monomer without a protease. Gag-Pol molecules can also unbind and rebind multiple times before successfully finding another Gag-Pol.

For the models with weaker and/or faster interactions, our simulations were long enough that all trajectories of that model (N_traj~60) resulted in a Gag-Pol dimerization event. We could thus construct the full FPT time distribution and reliably calculate the mean first passage times, τMFPT . The MFPTs displayed a remarkably clear functional dependence on both the ∆G_hex and k_a values as shown in Figure 4, for models where the sampling was complete. In contrast, for the most stable lattices with the slowest rates, the majority of the trajectories had not yet produced a dimerization event (Figure 4D, yellow bars), and thus the simulations do not report on the true MFPT. We therefore proposed a formula to fit the completed MFPT values, first by analogy to an MFPT model for bimolecular association, with an inverse dependence on k_a (see, e.g., Mishra and Johnson, 2021). Second, we empirically find that the τMFPT also has a power-law dependence on the K_D, τMFPT∝KDγ , or equivalently, τMFPT∝exp(γΔGkBT) . Our phenomenological formula thus had two fit parameters, the power-law exponent γ and a constant pre-factor (see Methods). After fitting, we find the approximate relationship:(1)τMFPT=7×10−5(ka2D/(4πR2))−1exp(−1.13ΔGhex/kBT)

where 7×10^–5 is a dimensionless fit parameter, R is the radius of the sphere, and our convention has ΔGhex<0. We see excellent agreement between our formula and our data (Figure 4C), which allows us to extrapolate the remaining models beyond the maximum simulation time of 20 s.

Importantly, in the models we have studied, the activation of a dimer can occur in well under a minute up through several minutes (Figure 4). For hexamer stabilities of –5.62 and –7.62k_BT, all rates support dimerization events at less than 10 s. For the more stable lattices of –9.62 and –11.62k_BT, only the medium and fast rates ensure an MFPT that is less than or comparable to (~50 s): an event occurring within 100 s of the start. Using our Equation 1, we can determine that for a moderate rate of 2.5×10^–2 nm²/μs, a ΔGhex more stable than –12k_BT will be slower than 100 s. For the slowest rate of 2.5×10^–3 nm²/μs, anything more stable than –9.4k_BT will be slower than 100 s. Our most stable lattices at the slowest rates take 10 min on average for an activation event. Our results thus quantify and predict how the kinetics and the stability of the lattice must be tuned to allow sufficiently fast dimerization events involving the 5% of Gag-Pol molecules carrying proteases.

When comparing lattices with 66% coverage vs 33% coverage, we see in some cases a minor slow-down in dimerization times, but the MFPT is overall much less sensitive than it is to the binding rates. With 33% coverage, the edge of the lattice does have a comparable size to the 66% lattice, but the ‘bulk’ interior is smaller, with more free space required to diffuse to a partner. However, most significantly, the concentration of Gag is smaller, and now with only 66 Gag-Pols (vs 125) present in the lattice, we see in some cases an increase in the time it takes for a pair to find one another (Figure 5A).

We tested two strengths for the dimerization free energy of ΔGdimer –11.62k_BT and –13.62k_BT, comparable to experimentally measured values of dimerization in solution (Datta et al., 2007). The MFPTs were overall relatively similar across both values, indicating that the timescales do not show the same sensitivity to ΔGdimer as ΔGhex over this range of free energies (Figure 5—figure supplement 1). This likely emerges because these dimer contacts are typically more stable than the hexamer. Thus, the hexamer unbinding events are more frequent and more likely to directly provoke the first activation events. Further, the hexamer has two binding sites, so more contacts in the lattice, and because hexamers nucleate stable cycles needed for higher order assembly, the frequency of hexamers vs incomplete hexamers are significantly more sensitive to ΔG than a single dimer bond. This result shows that breaking and formation of the hexamer contacts is important in driving Gag-Pol dimerization events, given that the MFPT shows clear sensitivity to these rates.

We similarly found minimal dependence of the MFPT on the diffusion constant. With a 10-fold slower diffusion constant for all membrane-bound Gag monomers, which effectively slows all lattices down by 10-fold, the MFPTs were not significantly slower (Figure 5B). This is not surprising given that the diffusional search along the membrane to find a new partner is not ultimately the rate-limiting step in the association process. The rates we report are intrinsic rates that control binding upon collision, whereas the macroscopic rates one measures through standard biochemistry experiments in the bulk are dependent on both this intrinsic rate and on diffusional times to collision (Collins and Kimball, 1949). Faster intrinsic rates are more diffusion-limited and produce binding that is more sensitive to diffusion (Yogurtcu and Johnson, 2015). However, given the small dimensions of the virion, traveling ~70 nm for example (the radius) takes on the order of milliseconds for monomers and small oligomers of Gag. Rough estimates of delay times for a binding event, using t~(k_a^2D*N_gagpol/SA)^–1 indicate that even for the fastest binding, it is on the order of a few milliseconds. The slowest timescale given all the rates is for the stable lattice, where dissociation has a timescale of ~7 s. Altogether, these timescales of individual steps show that the observed MFPTs are not merely controlled by the slowest single events, but by the need for multiple attempts of un- and rebinding to ensure a pair of Gag-Pols find one another. Our results further illustrate how the crowding due to the lattice on the surface can actually accelerate rebinding events compared to a freely diffusing pair when the lattice is unstable (Figure 5A), whereas for stronger Gag contacts the lattice will dramatically slow rebinding.

We find that the dynamics of our simulated lattices agrees with experimental measurements of binding within the lattice for parameters that exclude the most stable, slowest regimes. Experimental measurements within the Gag lattice of budded VLPs tracked the biochemical formation of a Gag dimer involving a population of Gag molecules tagged with a SNAP-tag (10–40%) and the same fraction of Gag molecules tagged with a HALO-tag (Saha et al., 2021). A covalently linked dimer was formed through addition of a HAXS8 linker at time 0, with one linker forming an irreversible bridge between a HALO and SNAP protein. Formation of this covalently linked dimer was quantified to reveal an initial rapid formation of dimers, followed by an increasing slower growth that reaches 42% dimer pairs formed for the 10% tagged populations. In our simulations, we thus performed a comparable ‘experiment’ given our trajectories (see Methods). We tracked the encounter between two populations of our Gag molecules that had been randomly tagged as either 10% HALO or 10% SNAP (Figure 6). We similarly found that the majority of the dimers formed rapidly, because they were already adjacent in the lattice when the covalent linker was introduced. The dynamics of the lattice then allowed a slow growth in additional dimers (Figure 6A). We calculated the fraction bound over the course of our 20 s simulations and used a simple extrapolation to define an upper bound on the number formed at 3 min (see Methods). For the least stable lattices, the upper bound is close to all dimers formed, over all three association rates, which is much higher than observed experimentally. For the most stable lattices in contrast, even when our model assumes maximal efficiency of the covalent linker, we extrapolate to an upper bound that is less than 42% dimers formed experimentally (Figure 6B). These results thus indicate that these lattices are too stabilized to support the dynamics observed in the Gag VLPs.

We also verified that these simulations are consistent with the trends expected as the population of tagged Gag monomers is increased. Indeed, we find, similar to experiment, that as a larger fraction of Gag monomers have tags, corresponding to a higher concentration of binding partners, we see a larger fraction of dimers being formed (Figure 6C).

We quantify the dynamics of the Gag lattice on fixed viewpoints on the spherical surface using number autocorrelation functions (ACFs), which report on correlations of collective motion that can emerge due to heterogeneity within the lattice (Figure 7) (Methods). We expect this heterogeneity due to our lattices all exhibiting a large component and smaller oligomers (Figure 2). We find that as the lattice becomes more stable and the bimodal separation of lattice sizes becomes more pronounced, the measured correlations in Gag copy numbers per quadrant increase in amplitude and slow in timescales, and that these dynamics are sensitive to slowing diffusion (Figure 7—figure supplement 1). All the ACFs will eventually asymptote to 1 at long delay times as the copy numbers become independent (Figure 7A).

In Figure 7, we show how the ACFs calculated from simulation show similar behavior to the ACFs measured from experiment. Each of the 8 quadrants of the spherical surface (4 on the top hemisphere, 4 on the bottom hemisphere) displays heterogeneity in the amplitude of correlations because some quadrants contain large lattice fragments, and others contain mostly empty space (Figure 7B). The same trend is observed in a single VLP measured using super-resolution microscopy imaging (Figure 7E). Our simulations further show that as the lattice is stabilized, the ACF increases in amplitude and decays more slowly, which is qualitatively the same as is observed in imaging of Gag lattices in budded VLPs that have been stabilized with a fixative (Saha and Saffarian, 2020; Figure 7C and F). We cannot quantitatively compare the ACFs, as the experiments produced ACFs with much higher amplitudes of correlations, and even with the background correlations divided out (Figure 7D), the experimental signal contained additional sources of correlation likely due to measurement noise. However, we were able to use our simulations to illustrate how sources of measurement noise in stochastic localization imaging experiments can produce increased correlations beyond the background. We specifically find that short-term blinking of the fluorophore does not appreciably change the ACF (Figure 7—figure supplement 1). However, we do see increased amplitude of correlations in the ACF if we introduce a distribution of activation probabilities for the fluorophores, mimicking the fact that the populations initially activated may have a higher probability of activation than those appearing at later times (Figure 7—figure supplement 2). Further, if we assume that the lattice is not perfectly centered with respect to the activating laser pulses, then Gag monomers that are initially ‘dark’ can diffuse into view and then have a probability of being activated (Figure 7—figure supplement 2). This increases fluctuations in both the background signal and the signal from the separate quadrants. Thus, the simulations improve interpretation of the experiment, and more vividly bring to life the lattice dynamics and heterogeneity.

Our model contains Gag and Gag-Pol monomers enclosed by a spherical membrane. The membrane contains binding sites for the Gag monomers. The Gag-Pol is structurally identical to the Gag but represents 5% of the total monomer population to track protease locations within the lattice. The model captures coarse structure of the Gag/Gag-Pol monomers as derived from a recent cryoET structure (Schur et al., 2016) of the immature lattice (Figure 1A). The key features of our rigid body models are the locations of the four binding sites/domains that mediate protein-protein interactions between a pair of Gag monomers and the Gag-membrane interaction. Each Gag/Gag-Pol contains a membrane binding site, a homo-dimerization site, and two distinct hexamer binding sites that support the front-to-back type of assembly needed to form a ring. When two molecules bind via these specific interaction sites, they adopt a pre-defined orientation relative to one another (Varga et al., 2020) that ensures the lattice will have the correct contacts, distances between proteins, and curvature (Figure 1 and Figure 1—source data 1). The Gag monomers bind to the membrane from the inside of the sphere, as would be necessary for budding, and we model this as a single binding interaction that captures stabilization from PI(4,5)P₂ binding and myristolyation (Ono et al., 2004; Saad et al., 2006). Each reactive site excludes volume from only its reactive partners at a distance σ. The dimer site reacts with another dimer site at a binding radius of σ=2.21 nm. The MA site binds to the membrane at σ=1 nm. The hexamer site 1 binds to hexamer site 2 at σ=0.42 nm (Figure 1B). Once reactive sites have bound to one another, they are no longer reactive and no longer exclude volume. Therefore, to maintain excluded volume between monomers throughout the simulation, we introduce an additional dummy reaction between the monomer centers-of-mass (COM). The COM sites exclude volume with a binding radius of σ=2.5 nm between all monomer pairs. This is necessary to prevent monomers from unphysically diffusing ‘through’ one another when their reactive sites are fully bound.

Computer simulations are performed using the NERDSS software (Varga et al., 2020). The software propagates particle-based and structure-resolved reaction-diffusion using the free-propagator reweighting algorithm (Johnson and Hummer, 2014). The membrane is treated as a fixed continuum surface that contains a population of specific lipid binding sites, or PI(4,5)P₂. We model these binding sites using an implicit lipid algorithm that replaces explicit diffusing lipid binding sites with a density field that will change with time as proteins bind or unbind from the membrane. Hence, PI(4,5)P₂ are assumed well mixed on the surface. This method reproduces the kinetics and equilibria as the explicit lipid method but is significantly more efficient (Figure 1—figure supplement 1; Fu et al., 2019). We use a time-step ∆t=0.2 μs. We validated the model kinetics as described in the next section. Software is open source here, https://github.com/mjohn218/NERDSS, and executable input files for the models are here, https://github.com/mjohn218/NERDSS/tree/master/sample_inputs/gagLatticeRemodeling.

We briefly describe here how the stochastic reaction-diffusion simulations work. Each protein or protein complex moves as a rigid body obeying rotational and translational diffusive dynamics using simple Brownian updates, for example xt+Δt=xt+2DΔtR, where D is the diffusion constant of the rigid body and R is a normally distributed random number with mean 0 and standard deviation 1. Each protein binding site is a point particle that can react with a site on another molecule to define the reaction network, as illustrated by the contacts in Figure 1. Reactions can occur upon collisions, with the probability that the reaction occurs evaluated using the Green’s function for a pair of diffusing sites, parameterized by an intrinsic reaction rate ka , a binding radius σ, and the sum of the diffusion constants of both species (Johnson and Hummer, 2014). This reaction probability is corrected for rigid body rotational motion (Johnson, 2018). For proteins that are restricted to the 2D membrane, they perform 2D association reactions with 2D rate constants (Yogurtcu and Johnson, 2015), which are derived from the 3D rate constants by dividing out a length-scale h that effectively captures the fluctuations of the proteins when on the membrane,(2)ka2D=ka3D/h

Proteins that do not react during a time-step undergo diffusion as a rigid complex, and excluded volume is maintained for all unbound reactive sites at their binding radius σ by rejecting and resampling displacements that result in overlap. All binding events are reversible, with dissociation events parameterized by intrinsic rates kb that are sampled as Poisson processes. We have that for each reaction, KD=kbka , and for the corresponding 2D reaction, we assume the unbinding rates are unchanged, and thus KD2D=hKD3D .

Binding interactions are dependent on collisions between sites at the binding radius σ and are not orientation dependent. Orientations are thus enforced after an association event occurs by ‘snapping’ components into place. Association events are rejected if they generate steric overlap between components of two complexes. Steric overlap is determined using a distance threshold, where here if the distance between molecule COM is less than 2.3 nm, we reject due to overlap. They are rejected if they generate large displacements due to rotation and translation into the proper orientation, using a scaling of the expected diffusive displacement of 10. Defects ultimately emerge in the lattice because a hexagonal lattice cannot perfectly tile a spherical surface by the Euler polyhedron formula. These defects result in contacts that are not perfectly aligned (Figure 2); if the contacts are within a short cutoff distance of 1.5σ, they can still form a bond to stabilize the local order, otherwise they are left unbound, weakening the local order.

We estimate translational (D) and rotational (D_R) diffusion coefficients from the Einstein-Stokes equations, assuming a higher viscosity for an in vivo process. We define for a Gag in solution: D_Gag(D_Gag-Pol)=10 μm²/s, D_R,Gag(D_R,Gag-Pol)=0.01 rad²/μs. Membrane D_lipid = 0.2 μm²/s. Diffusion slows as complexes grow, consistent with a growing hydrodynamic radius and quantified by the Einstein-Stokes equation (Varga et al., 2020). Hence, a single protein on the membrane diffuses at 1.96×10^–1 μm²/s, and a membrane bound complex containing 1000 proteins diffuses at 1.96×10^–4 μm²/s.

We studied lattice dynamics at several strengths defining the free energy ΔGhex of the hexamerization interaction, at −5.62kBT, −7.62kBT, −9.62kBT, −11.62kBT, where kB is Boltzmann’s constant and T is the temperature. The Gag and Gag-Pol have identical energetic and kinetic parameters to one another during all remodeling simulations. We specified the dimerization free energy ΔGdimer at −11.62kBT and −13.62kBT, which straddles the stability of the measured solution KD of 5.5 μM or −12.1kBT (Datta et al., 2007). Additional stabilization of the dimer interaction can accompany conformational changes (Datta et al., 2011), which could drive stronger Gag-Gag binding within the lattice (Datta et al., 2007). Given the ∆G values (ΔG=Gbound−Gunb) and using, KD=c0exp⁡(ΔGkBT) , where c0 is the standard state concentration (1 M), we further selected a set of on- and off-rates at each free energy, where we used both hexamer and dimer intrinsic binding rates k_a^2D of 2.5×10^–3 nm²/μs, 2.5×10^–2 nm²/μs, and 2.7×10^–1 nm²/μs. Off-rates are constrained by ∆G via KD=kb/ka3D,ka2D=ka3D/h . For the Gag-membrane interaction, h=2 nm, for the Gag-Gag interactions, h=10 nm, comparable to the size of the Gag monomer.

Our model allows for a strain energy in the formation of closed polygons-like hexagons. We set that energy ΔGstrain here to +2.3kBT for all models, meaning that the stability of any closed hexagon within the lattice is slightly lower compared to 6 ideal bonds (i.e. for ΔGhex=−11.62kBT it is 5.8 bonds) but still much more stable than a linear arrangement of six Gag monomers which has only 5 bonds. This is an entropic penalty to forming closed cycles which require the final subunit to fit into the 5-mer structure and form 2 bonds simultaneously. This could mechanistically result from compressed or stretched arrangement of subunits in the hexameric cycles on the curved membrane, compared to a more favorable spacing in solution where no forces from the membrane exist. This penalty only affects the lifetimes of the hexamer cycles. When a hexamer closes to form a cycle, it forms 2 bonds and thus has a stability of KD,cycle=c0exp⁡(2ΔGhexkBT)exp⁡(ΔGstrainkBT)=KDexp⁡(ΔGhex+ΔGstrainkBT), where ΔGhex<0 and a penalizing ΔGstrain is >0. We perform association reactions with the same forward rate, which means that kb,cycle=kaKD,cycle=kbexp⁡(ΔGhex+ΔGstrainkBT). We do not apply this strain penalty to dimer bonds that can also end up in higher-order cycles, thus assuming that they can accommodate spacing or small structural rearrangements without any free energy cost. We ran a set of comparison simulations where ΔGstrain=0, to illustrate how it can impact the structures of the weaker lattices. Quantitatively, setting ΔGstrain to 0 gives the hexamer cycles a lifetime that is 10-fold longer. For ΔGhex=−11.6kBT, the hexamer lifetime thus increases from 1380s to 13,800 s when k_a^3D=1.5×10⁵ M^–1s^–1, but these are both dramatically slower than a single hexamer bond which has a lifetime of 0.74 s. For the weakest lattice, however, the hexamer cycle is only 4× more long-lived than a single bond, so the strain penalty is more impactful. With additional dimer interactions stabilizing subunits in the lattice, however, hexamer lifetime increases further.

We validated the kinetics and equilibrium of our model as it assembled on the membrane when we set the hexamer rates to 0, so it formed purely dimers (Figure 1—figure supplement 1), and when we set the dimer rates to 0, so it formed purely hexamers (Figure 1—figure supplement 2). The observed kinetics and equilibria were compared to solutions solved using the corresponding system of non-spatial rate equations, showing very good agreement with apparent intrinsic rates that systematically accounted for excluded volume and the criteria used for accepting association events (Figure 1—figure supplement 1). Thus, all of the rates and free energies reported in the paper agree with the kinetics and equilibria observed and expected for the sets of binding interactions that make up the full lattice system.

The HIV lattice is composed of Gag and Gag-Pol bound to the inner leaflet of the lipid membrane. To study the remodeling dynamics, we must construct the initial configurations where the lattice is assembled such that it has a specific coverage of the surface (67% or 33%), and is linked to the membrane via lipid binding. We define the membrane sphere of radius 67 nm to represent the membrane surface (Sundquist and Kräusslich, 2012). PI(4,5)P₂ is populated on the membrane surface at a concentration 0.07 nm^–2, or 4000 copies, which exceeds the number of Gag monomers, meaning there is always a pool of free PI(4,5)P₂ available for (re)binding.

Assembling the Gag monomers into a single spherical lattice is non-trivial due to the size of the lattice. Because the lattice is so large, requiring N~2400 monomers at 67% coverage, it is very difficult for a single nucleated lattice to complete growth (which scales approximately with N) before another lattice nucleates. These multiple intermediate fragments do not readily combine. In a recent study we quantified how titrating in monomers instead of trying to assemble from the bulk can dramatically improve assembly yield (Qian et al., 2023). Therefore, here we titrate in the Gag and Gag-Pol monomers at a rate of 6×10^–5 M/s and 3×10^–6 M/s respectively, which can ensure a ratio of Gag:Gag-Pol of ~20:1, consistent with experiment (Sundquist and Kräusslich, 2012; Garcia-Miranda et al., 2016). Gag(Gag-Pol) molecules can bind in solution (3D), to the membrane (3D to 2D), and when on the membrane (2D). In one set of assembly simulations we set binding rates between Gag-Pol and Gag-Pol pairs to 0 to try and suppress the ‘activation’ events that could therefore occur during assembly (Figure 2). While the titration of the monomers reduced multiple nucleation events for the membrane system, we found that the easiest and most efficient way to form a single lattice was by assembling the structure fully in solution, in a volume of (250 nm)³. We then put the assembled single lattice into a spherical system by linking this structure to the membrane using one PI(4,5)P₂ attachment per monomer. The Gag rates of dimerization and hexamerization were both set to 6×10⁶ M^–1s^–1.For the lattices studied below, we made binding events irreversible, as it improved the growth of single lattices. For comparison, we also ran a few assembly simulations where the binding was reversible, using ΔGhex=−11kBT and ΔGdimer=−13kBT, k_off =100 s^–1 and 13.6 s^–1, respectively. These reversible binding simulations also used titration, and although they often nucleated two structures, we could keep adding monomers until at least one lattice reached our target size. Because the hexamer and dimer rates are identical during the assembly process, we do not see selection for only complete dimers along the lattice periphery, as is observed in the cryoET maps (Tan et al., 2021). To recover this feature, we would instead need to assemble the lattice under more native-like conditions where the dimer is more rapidly and stably formed compared to the hexamer contact. We generated 16 initial configurations for each coverage area (67% and 33%). Some initial configurations are shown in Figure 2.

For each initial configuration we have generated, we perform six independent trajectories. See Video 1 for one trajectory. We perform these 96 simulations for each set of model parameters to generate statistics both within and across initial configurations. For some simulations, fragments of the lattice become sterically overlapped with one another, due to the high density and the time-step size. While this could be eliminated by lowering the time-step, we instead keep the more efficient time-step, and discard these simulation traces which produce overlap. We finally analyze 60 remodeling traces for each parameter set. All the simulation parameters are listed in Table 1. The number of monomers is fixed for each simulation by the initial configuration, so that only binding, unbinding, and diffusion can occur throughout the simulation. During lattice construction, in one set of simulation we set all Gag-Pol to Gag-Pol binding interactions to zero (Figure 2B). Now for the remodeling dynamics, we allow all interactions involving Gag-Pol, and at rates that are identical to those involving Gag, meaning there is no difference between the types except for in their label. The Gag/Gag-Pol molecules are allowed to diffuse on the membrane, where they can unbind from a molecule and rebind to another with the specified binding rates. Each monomer can also unbind and rebind to the membrane lipids. However, dissociation to solution is extremely rare, as it requires that all Gag monomers in an assembled complex unbind from their lipid before any of the sites rebind. In Figure 1—figure supplement 4, we confirm that even for the most unstable lattice that produces small mobile fragments, the dynamics are very similar when the lipid binding is modeled using implicit or explicit binding sites.

Our primary observable is how long it will take for the first dimerization event between a pair Gag-Pols Our simulations are stochastic and thus this is an FPT measurement (Iyer-Biswas and Zilman, 2016). The ‘clock’ is started from the initialized lattices as shown in Figure 2. We note that two Gag-Pols have a chance to be adjacent at the initial configuration (Figure 2), but we ignore these events since dimerization of Gag-Pol before viral release has been experimentally shown to result in loss of Pol components from the virions (Bendjennat and Saffarian, 2016).

Our primary observable is how long it will take for the first dimerization event between a pair Gag-Pols. Our simulations are stochastic and thus this is a first-passage time measurement (Iyer-Biswas and Zilman, 2016). The ‘clock’ is started from the initialized lattices shown in Figure 2. We note that two Gag-Pols have a chance to be adjacent at the initial configuration (Figure 2), but we ignore these events since dimerization of Gag-Pol before viral release has been experimentally shown to result in loss of Pol components from the virions (Bendjennat and Saffarian, 2016).

Recent measurements on Gag VLPs quantified dimerization times between sub-populations of tagged Gag molecules within the immature lattice (Saha et al., 2021). Dimerization events were identifiable because one sub-population of Gag monomers carried a HALO-tag (a protein that fuses to a target of interest, here Gag), and another carried a SNAP-tag. The addition of a linker HAXS8 produced a covalent linkage which we will call HALO-link-SNAP. The concentration of these HALO-link-SNAP structures was then quantified vs time. We therefore reproduced this experiment via analysis of our simulation trajectories. We defined a population of our Gag monomers randomly selected to have a ‘SNAP-tag’ and a population randomly selected to have a ‘HALO-tag’. For each trajectory, the tagged populations of each were either 5%, 10%, 20%, or 40% of the total monomers, to match the experimental measurements (Saha et al., 2021). We then monitored the number of dimerization events that occurred as a function of time. A dimerization event required that a monomer with a SNAP-tag and a monomer with a HALO-tag encountered one another at a distance less than 3 nm (Erhart et al., 2013), where one and only one of these partners must have the covalent linker attached. Three nm cutoff distance is comparable to the molecular length-scales of the two protein tags with the linker between them (Erhart et al., 2013). We randomly selected half of the population of SNAP-Gags to have a linker attached, and half of the population of HALO-Gags to have a linker attached, and thus some encounters between a SNAP- and HALO-Gag were not productive if 0 or 2 linkers were present. Ultimately, however, all dimers could be formed given the symmetric populations containing linkers. These binding events were irreversible, consistent with a covalent bond formed.

This model assumes that the arrival of the linker to the inside of the virion is relatively rapid. The permeability coefficient of the linker when exposed to the membrane enclosed Gag lattice is approximately 0.0004 nm/μs (Erhart et al., 2013), and assuming a membrane thickness of ~5 nm, the diffusion across the membrane occurs at ~0.002 nm²/μs. To test the role of linker permeability, we solved the diffusion equation for a 1 μM concentration of linker molecules diffusing into a sphere of radius R=67 nm, which mimics the experiments. Within 100 ms, the concentration of the linker at 60 nm (close to the Gag-tagged end) has already reached 0.8 μM. Hence, although there is some delay following addition of the linker, it is much less than the time (20 s to 3 min) over which most of the dimerization occurs. The model also assumes that the linker does not saturate all HALO and SNAP molecules independently, which would prevent any dimers forming. The rates of binding of SNAP and HALO to the linker HAXS8 are 3×10⁴ and 3×10⁶, respectively (Erhart et al., 2013). We solved a system of ordinary differential equations for binding of HALO and SNAP to a linker given these rates. The HALO and SNAP concentrations were controlled by the size of the virions with 250 of each present (10%), and the linker concentration was 1 μM, which was found experimentally to ensure high dimerization success (Saha and Saffarian, 2020). Although the linker binds more rapidly to HALO, there is still plenty of time for the HALO-linked molecules to bind to a free SNAP before all the sites are occupied by linkers, as the copy numbers of linkers in the volume are low. In particular, if the HALO and SNAP tags are adjacent in the lattice, the SNAP is much more likely to bind the adjacent HALO-linker than a free linker.

Since our simulations are ~20 s, we did a linear fit of the last second of the dimer forming kinetics to extrapolate the dimer copies at 3 min, which can be used for comparison with the experiment. This extrapolation therefore assumes that dimer formation does not slow down, which it almost certainly does. All of our assumptions contribute to the maximal possible dimerization efficiency, and thus our observables provide an upper bound on the expected number of HALO-link-SNAP dimers.

Recent experiments also measured an ACF of immature lattice dynamics using time-resolved super-resolution microscopy. These experiments on Gag VLPs used interferometric photoactivated localization microscopy (iPALM) to track collective motion of the Gag lattice by stochastically localizing individual monomers to precise locations in the lattice over several minutes (Saha and Saffarian, 2020). To mimic the experimentally extracted observable, we counted the number of Gag monomers found on each fixed 1/8 of the sphere surface. The copy numbers within each of the 8 quadrants vary due to diffusion of the lattice, while the total copy numbers on the surface is fixed. The copy numbers at a time point t are denoted by Gt , and thus the ACF for each quadrant is given by:(4)ACFτ=Gt+τGtGt+τGt

These ACF measurements are comparable to fluorescence correlation spectroscopy measurements (Wohland et al., 2001). For comparison, the ACF for Gt across the full sphere surface is 1 at all times, because there is no change in total copy numbers. At long times, when the counts are uncorrelated, this function will go to 1, because Gt+τGt→Gt+τGt. Furthermore, if the copies are all well mixed across the surface, then we expect minimal deviations from 1 across all times given these relatively large viewing regions (1/8 of the surface), because there is no source of correlation between copies if they are well mixed. As τ→0, the deviation of this ACF from 1 reports on the variance of the fluctuations in the numbers per patch relative to the mean, or the coefficient of variation (CV) squared: ACFτ=0=Gt2-Gt2Gt2+1=σ2μ2+1. We calculated the number correlation for each quadrant of one simulation trace. We then took the average of all 60 traces for each parameter set. We also used an ensemble averaging approach, where we calculated the copy number correlations and means across multiple quadrants and trajectories before averaging to get the numerator and denominator of Equation 4. This method provides more statistics on longer time delays, and is based on assuming that all trajectories are sampling from the same equilibrium distribution. The trends are the same as those we report but shifted up to slightly higher amplitudes before decaying to 1. We note that our ACFs are not truly reporting on equilibrium fluctuations, as we show below that there is clearly some time-dependent changes to the lattice structure that is not reversible due to fragmenting along the edge compared to the initial structures. However, we compared the ACFs calculated for the first half of our simulations vs the last half, for example, and all of the same trends are preserved.

In addition to directly calculating this number autocorrelation, we also sampled it using a stochastic localization approach that directly mimics the experimental measurement (Saha and Saffarian, 2020). For this approach, for each time point we ‘activate’ a single Gag monomer across the full lattice with a probability p_act. So for each frame, either 1 or 0 Gag monomers is visible. We identify the quadrant for that monomer, and thus each quadrant produces a sequence of 1s and 0s. After a monomer is activated, it is then bleached, and cannot be localized again. The sequence of localizations is then used to calculate the same ACF, where we use a binning method (Wohland et al., 2001), as in the experimental analysis, to improve statistics on the signal at larger time delays. The agreement between the stochastic measurement of the ACF and the direct measurement of the copy numbers ACF are excellent (see Fig 7). We use this stochastic localization method so that we can introduce additional sources of correlation to our measurement of the simulated lattice dynamics, since these measurement artifacts can appear in the real experimental system.

The time-resolved microscopy (iPALM) experiments to characterize lattice dynamics in VLPs were previously described and published (Saha and Saffarian, 2020). We describe the analysis of these stochastic localization experiments here because we focus on analyzing a shorter part of the measurement. We analyzed only the first 500 s of the measurement (5000 frames), because after that time the laser intensity was changed. Based on data collected on 25 VLPs, we analyzed only the VLPs where they reported a large enough fraction of localization events to indicate a reliable measurement, so we included only VLPs where >75% of the quadrants had more than 250 localizations, leaving 11 VLPs. We used the same algorithm (Wohland et al., 2001) as applied to the simulation data to quantify the ACF from the time-dependent sequences of localization events (typically a series of 1s and 0s, with occasionally 2 events per frame). For each of the 8 quadrants, we effectively measure the copies of monomer per quadrant: n₁(t), n₂(t),... n₈(t). The total copies are then N(t)= n₁(t)+n₂(t)+... +n₈(t). The ACF for any single quadrant is given by ACFτ=nit+τnitnit+τnit . For the total surface, ACFτ=Nt+τNtNt+τNt , which we denote as the background signal, as the full surface was visualized once per experiment, with localization events then assigned to quadrants. This background signal reports on fluctuations in the total copy numbers of Gag on the surface, which we would expect to be 1 given a perfect measurement, but which was always higher than this due to measurement noise. To remove this effect of total copy number variations, and instead focus on the local fluctuations in concentrations per quadrant, we would like to report a corrected ACFcorr(τ)=⟨ni(t+τ)/N(t+τ)ni(t)/N(t)⟩⟨ni(t+τ)/N(t+τ)⟩⟨ni(t)/N(t)⟩ . However, we do not have access to the relative concentrations nit/Nt at each time-step from experiment. A reasonable approximation is to assume that we can separate the average behavior of nit and Nt , such that ACFcorr(τ)≈⟨ni(t+τ)ni(t)⟩/⟨N(t+τ)N(t)⟩⟨ni(t+τ)⟩⟨ni(t)⟩/⟨N(t+τ)⟩⟨N(t)⟩, which is equivalent to dividing out the background ACF from the signal of each quadrant. This background-corrected ACF reproduces the exact ACF when the total copy numbers are constant. We then averaged these signals across the VLPs. We performed the same analysis on the 25 VLPs that had been modified by a fixative, first filtering out the VLPs that had too few localization measurements (leaving 13 VLPs), and then otherwise proceeding with the analysis in an identical fashion.