NERDSS: A Nonequilibrium Simulator for Multibody Self-Assembly at the Cellular Scale

Abstract

Currently, a significant barrier to building predictive models of cellular self-assembly processes is that molecular models cannot capture minutes-long dynamics that couple distinct components with active processes, whereas reaction-diffusion models cannot capture structures of molecular assembly. Here, we introduce the nonequilibrium reaction-diffusion self-assembly simulator (NERDSS), which addresses this spatiotemporal resolution gap. NERDSS integrates efficient reaction-diffusion algorithms into generalized software that operates on user-defined molecules through diffusion, binding and orientation, unbinding, chemical transformations, and spatial localization. By connecting the fast processes of binding with the slow timescales of large-scale assembly, NERDSS integrates molecular resolution with reversible formation of ordered, multisubunit complexes. NERDSS encodes models using rule-based formatting languages to facilitate model portability, usability, and reproducibility. Applying NERDSS to steps in clathrin-mediated endocytosis, we design multicomponent systems that can form lattices in solution or on the membrane, and we predict how stochastic but localized dephosphorylation of membrane lipids can drive lattice disassembly. The NERDSS simulations reveal the spatial constraints on lattice growth and the role of membrane localization and cooperativity in nucleating assembly. By modeling viral lattice assembly and recapitulating oscillations in protein expression levels for a circadian clock model, we illustrate the adaptability of NERDSS. NERDSS simulates user-defined assembly models that were previously inaccessible to existing software tools, with broad applications to predicting self-assembly in vivo and designing high-yield assemblies in vitro.

Significance

A major roadblock in quantitative modeling of cell biology is the need to resolve fast processes over long timescales and individual proteins over large length scales. This is particularly evident in self-assembly processes such as clathrin-mediated endocytosis and viral assembly, which proceed over minutes, building from often weak and short-lived binding events. The NERDSS software presented here uses the reaction-diffusion model to simulate such self-assembly processes, with broad applications to biological processes as they occur in the nonequilibrium cell.

Introduction

Watching the dynamics of individual molecules in the cell as they function as part of a collective is now possible thanks to revolutions in live-cell microscopy. Despite technical advances in mapping dynamics onto nanoscale structure formation in the cell (1), resolution is still limited by events that are too fast and combine too many distinct components. Computer simulations offer the promise of reproducing dynamics using models designed from the underlying physics and mechanics, providing high-resolution spatial and temporal predictions and exquisite control over components and their interactions. Current tools for capturing cell-scale complexity are a growing companion to cell biology. For example, fields such as cell-signaling (2), developmental biology (3,4), and systems biology (5) have benefitted from a variety of spatial (6,7) and nonspatial tools (8,9) in which interactions and reactions are modeled as events parameterized by rate constants. However, many cell-scale processes involve self-assembly, which is a challenge for modeling because it spans similarly long length and timescales as biochemical signals while also depending fundamentally on molecular structural geometry. Although rate-based approaches have been shown to be highly insightful for modeling self-assembly, previous models lack explicit spatial resolution (8,10, 11, 12), apply only to small systems (13,14), or include potentials that prevent quantitative comparison to experiment (15). Here, we present the nonequilibrium reaction-diffusion self-assembly simulator (NERDSS), a higher-resolution rate-based software tool with the addition of user-specified, coarse molecular structure to enable the study of self-assembly at the cell scale.

NERDSS was developed to build self-assembly into the reaction-diffusion (RD) model because of the strength of RD in simulating relatively slow, nonequilibrium dynamics. The name of the software may sound somewhat redundant (particularly NERD). The “nonequilibrium” is included in the name to emphasize how self-assembly can be simulated here in generalized, nonequilibrium systems such as the cell. Additionally, although the RD model is commonly applied to studying systems not at thermodynamic equilibrium (16), it is nonetheless useful for quantifying the pathways to and composition of equilibrium for complex multicomponent systems. The standard computational approach for studying self-assembly, using coarse-grained molecular modeling (17, 18, 19), provides access to these pathways and equilibria with more physically detailed models, in which interactions emerge because of distance-dependent energy functions rather than the rate-controlled events of RD models. A coarse molecular model thus naturally can accommodate a range of structures or defects (17,19,20) and capture emergent cooperativity, strain, and localization of components (21,22). However, they are still relatively limited in the length and timescales of dynamics they can access, which does not extend to the cell scale. They are parameterized by potential energy functions, and converting these to the binding free energies (e.g., as done here (21,23)) or rate constants (24, 25, 26) for comparison to experiment is nontrivial. Lastly, chemical reactions require covalent bond breaking that is not typically accessible in molecular modeling, preventing systematic and transferable methods for involving enzymatic or ATP-driven reactions ubiquitous in cells. These models are unable to simulate in vivo cell signaling, cytoskeletal dynamics, or clathrin-mediated endocytosis, for example. Our NERDSS software addresses this substantial application gap, and although it lacks the detail of energy-function based models, it is able to uniquely preserve important features of molecular assembly.

NERDSS overcomes technical challenges with adding structure to RD and allowing large, reversible complexes to form in a user-friendly and widely applicable way. The major challenges for applying RD to self-assembly are expanding to multisite, volume-excluding species and capturing orientation-dependent interactions for arbitrary species. For continuum partial differential equation (PDE) approaches (6,7) to RD and lattice-based reaction-diffusion master equation (RDME) approaches (27, 28, 29), species have no individual coordinates but are well mixed in the infinitesimal or finite grid spacings, respectively, preventing the resolution of structural assemblies. Capturing assembly kinetics is possible in these methods for complexes with uniquely defined subunits, with recent RDME-based studies of ribosome formation (30) and essential steps in spliceosome assembly (31). Although single-particle RD is considerably more computationally expensive than PDE or RDME, it does track individual coordinates (32, 33, 34, 35, 36, 37, 38, 39), often with excluded volume, providing a starting point for multisite species and structure resolution. Recent methods have generated multisite species by linking them using spherically symmetric potentials (14,40), producing simulations that can capture effects of clustering (41) and polymer assembly (40). However, without orientation-dependent interactions, well-defined structural geometries such as lattices and spherical shells will not form. Brownian dynamics methods have been successfully used to capture orientation-dependent, rate-controlled association but are limited to small systems (42,43). Here, we use our recent algorithm for structure-resolved RD of rigid bodies, which includes and accounts for effects of rotational as well as translational diffusion on binding reactions (44). Instead of using potentials, orientational constraints are applied after binding events to maintain accurate solutions to the RD equations of motion and direct comparison to experimental rates. With previous application of this algorithm to clathrin lattice assembly in solution (44), NERDSS generalizes these routines for orienting molecules to apply here to arbitrary user-defined systems, including on the membrane, which can act as a potent driver of self-assembly via dimensionality reduction (45). This approach retains the flexibility and the adaptability of the method to new molecules and structures because binding is parameterized by rates and not customized interaction potentials.

A critical feature of generalized rate-based software for usability, model portability, and construction of conditional and cooperative interactions is the adoption of well-defined formatting languages for model encoding. Rule-based modeling (46), developed for nonspatial models, supports descriptions of species as multisite molecules in which each site can exist in distinct states. Interactions or binding rates can be coupled within species to capture effects of phosphorylation, for example, on subsequent binding events. Formatting languages such as BioNetGen Language (46,47) and Kappa (48,49) implement rule-based modeling, which precludes the need to define and track all possible species and avoids the issue of combinatorial complexity (50). The rule-based format is now available in efficient nonspatial methods such as RuleBender (51) and NFSim (8) and spatial methods such as Virtual Cell (52) and Smoldyn (34), facilitating model comparison and portability. Rule-based encoding translates quite naturally to NERDSS because of the spatial specification of each site. We validate our rule-based encoding here by simulating a model of a circadian clock (53) (and other simple models), compared with the same model simulated in Virtual Cell (6).

To illustrate the capabilities of NERDSS as a spatial self-assembly simulator, we apply it here to steps in clathrin-mediated endocytosis (CME) and virion assembly. These examples highlight several challenges and opportunities for modeling approaches, and both assembly processes have been studied using coarse molecular modeling simulations (19,54, 55, 56, 57). CME is an essential pathway used by all eukaryotes for transport across the plasma membrane. It is a stochastic assembly process that depends on membrane mechanics (1,58), enzymatic reactions (59, 60, 61), and the stoichiometry of dozens of distinct components (59,62, 63, 64), making it a complex system that can benefit from integrated physics-based simulations. Spatial simulations of clathrin, a 600 kDa trimeric protein, assembling in solution (44,54,56,65) and on the membrane (via recruitment by adaptor proteins) (19) have determined assembly pathways, energetic and structural determinants for assembly, and dependence on links to the membrane. The clathrin models developed here for NERDSS illustrate how rate-based models can be adapted and extended to additional components without requiring the substantial expertise of energy-function-based methods (19,66,67) and without sacrificing spatial resolution (45,68, 69, 70). To enable nonequilibrium simulations of immature viral lattice formation in vivo, we introduce here a model of the retroviral Gag protein, which assemble a spherical lattice (71), an essential component of the HIV infection and maturation cycle (72). Viral capsid assembly is widely studied using coarse molecular modeling approaches (see (17)), but applications to studying assembly of the precursor lattices that occur within the host cell are more limited (55). In this study, we focus on describing the technical features of the models necessary to produce realistic assemblies, leaving much of the biological implications for future work.

In summary, NERDSS offers a distinct tool that uses the RD model to simulate self-assembly at the cell scale, thus allowing for space- and time-dependent dynamics and kinetics that are immediately comparable to experiment by a broad user base. Below, we first describe the operation of the software and the features we introduced as necessary to move beyond existing tools (Fig. 1). We provide multiple forms of validations, including against fundamental reactions and nonassembly problems such as a circadian clock model, accessible to most RD software. We then present multiple self-assembly applications, with input files provided in the software repository. Models of clathrin lattice assembly are chosen to highlight distinct features of models that can be used to tune the dynamics and stability of the observed structures. We simulate clathrin assembly driven by membrane localization and show how introducing enzymes to the system to control the lipid populations on the membrane drives disassembly. These clathrin models thus capture relaxation from out-of-equilibrium starting states to an equilibrium steady state, with or without active energy consumption. We illustrate the model design process for the self-assembly of Gag retroviral protein monomers into an immature lattice in solution, in which monomers are titrated into solution and have limited lifetimes when unbound. Finally, we discuss the current limitations and most promising future advancements of NERDSS software for realistic cell-scale dynamics.

Figure 1.

NERDSS software overview. (a) NERDSS requires additional inputs relative to other rate-based models, highlighted in yellow boxes, with each species modeled as a rigid body with coordinates for a COM (red) and its discrete binding sites (blue) and a binding orientation for a pair of bound species. These orientation parameters include a binding radius, σ, shown as a purple line between two sites; two angles, θ₁ and θ₂; and three dihedral angles, φ₁, φ₂, and ω (see Supporting Materials and Methods). NERDSS uses free-propagator reweighting (FPR) algorithms to solve the RD model for association and dissociation over time. These algorithms were previously accessible in open source code but without rule-based models and coupling to other reaction types. In addition to coordinates of all species in the simulation as a function of time, NERDSS can track a variety of variables, including species copy numbers, which are formatted for analysis and visualization. Trajectories are output in one of two standard formats, XYZ or PDB, which can be visualized in software such as VMD (90) or Ovito (91). (b) An algorithmic flowchart highlights where modifications to time and/or space dependent rate-based models (gray) are needed because of the addition of molecular structure and assembly (yellow boxes) (Methods). Reactions can be zeroth order: creation; first order: death, transformations, dissociations; second order: binding. For each time step, every molecule either reacts or diffuses. To see this figure in color, go online.

Methods

Implementation

NERDSS is written in ANSI/ISO standard C++11 and is available on Linux and macOS. The accompanying GUI is implemented in Java. The source code for both can be found at https://github.com/mjohn218/NERDSS and is provided under the GNU general public license. A user guide and simulation example systems can be found on the software page. Input files are formatted according to BioNetGen Language (47), such as are used in RuleBender (51) and the NFSim software (8) and is compatible with Virtual Cell (6). Additional features are necessary, however, because of the spatial and structural details used in NERDSS.

Reactions

NERDSS supports the three primary types of reactions: zeroth, first, and second order. The rates of these reactions, other than creation, can depend on the states and/or pre-existing interactions of any participant species’ sites.

Zeroth-order reactions

De novo particle creation reactions are treated as a Poisson process. No events occur if a uniform random number (URN) fulfills URN < exp(−λ), where λ = k₀VΔt, k₀ is the creation rate input in units of M/s, V is the simulation volume (converted from nm³ to units of M⁻¹ by 0.602), and Δt is the timestep. Therefore, N events occur based on $\sum _{n=0}^{N-1}\left(\text{exp}\left(-\lambda \right){\lambda }^n/n!\right)<URN\le \sum _{n=0}^N\left(\text{exp}\left(-\lambda \right){\lambda }^n/n!\right).$. Created particles are placed within the simulation volume with random coordinates. Zeroth-order reactions can be used to titrate species into the simulation volume.

First-order

First-order reactions are treated as Poisson processes, with a reaction probability per each site of p₁(Δt) = 1 − exp(−λ), where λ = k₁Δt, k₁ is the microscopic rate in units of s⁻¹, and Δt is the time step. The reaction rates can be specified either as microscopic rates or as macroscopic rates. Only one such event can occur per time step per molecule. Four types of first-order reactions are supported:

• 1)Creation: create a copy of a particle B from another particle A, in which sites required such as (a) are indicated within the parentheses—examples are transcription or translation, e.g., A(a) → A(a) + B(b).
• 2)Destruction: destroy a molecule and all other molecules in its complex. If the sites (a) are specified as unbound, it will only destroy unbound molecules, e.g., A(a) → NULL.
• 3)State change: change a state of a site on a molecule, e.g., A(a∼p) → A(a∼u).
• 4)Dissociation: remove an interaction between the sites of two particles, written as the conjugate back reaction of an association reaction. To retain detailed balance (32), the particles are left such that the dissociating sites are at their binding radius, e.g., A(a!1).B(b!1) → A(a) + B(b).

Second-order reactions

Bimolecular reactions are applied to specific sites between two proteins. Association events can be conditional on the state of the site, e.g., occurring only when the binding site is phosphorylated, or only if the protein is bound through another site as well.

For the free propagator reweighting (FPR) algorithm (32,44,73) used in NERDSS, reaction probabilities are invariant with respect to orientation. Therefore, the reacting particles are “snapped” into place in a predefined geometry to prevent arbitrary structures from forming, with sites always placed at the binding radius σ from one another. Thus, the rigid assemblies that form have strict structures they can form; they do not predict emergent structures. The geometry is defined by a set of vectors and five angles between and within each particle (Fig. 1). If angles and/or orientations are undefined, sites bind to a separation of σ at the orientation they were when the event occurred. Both molecules are translated and rotated into place based on their relative translational and rotational diffusion constants. Thus, smaller complexes will move more to orient, and complexes restricted to the membrane do not rotate out of their membrane-localized orientation. Detailed definitions of these vectors and angles can be found in the Supporting Materials and Methods.

Reaction rates can be defined either as macroscopic rates k_on (i.e., rates measured experimentally and dependent on diffusion to contact and energetic barriers), in units of μM⁻¹s⁻¹, or as microscopic or intrinsic rates k_a, in units of nm³/μs (converts to μM⁻¹s⁻¹ with multiplication by 0.602). All rates are inputted as three-dimensional (3D) values, where K_D = k_b/k_a = k_off/k_on, where k_b and k_off are the corresponding unbinding rates in units s⁻¹. For example, reaction rates in two dimensions (2D) use the 3D value divided by a length scale that by default is set to 2σ (units of nm) but that can be independently specified per reaction using an input parameter. A reaction is identified as 2D when it involves two species that have no diffusion in z (e.g., lipids) or between sites on two complexes that are localized to 2D (D_z = 0). Probabilities of each reaction are calculated with the FPR method as previously described and are parameterized by the intrinsic reaction rate k_a, the net diffusion coefficient of the reactants D_tot, and the binding radius σ. Conversions between macroscopic rates and the rates needed to calculate the reaction probabilities for each reaction type are performed by the software (see Table S1). Self-binding between molecules such as clathrin with repeated identical sites requires distinctions between binding sites with the same versus distinct labels (see Supporting Materials and Methods). Two types of bimolecular reactions are supported:

• 1)Association: form an interaction between two sites of two molecules, which, if the association is reversible, has a conjugate first-order dissociation reaction. The resulting complex is then treated as one unit for future propagation, e.g., A(a) + B(b) → A(a!1).B(b!1)
• 2)State change: change the state of a site on a particle, facilitated by another particle, such as phosphorylation by a kinase. This is thus a composite reaction that combines a bimolecular reaction with an automatic first-order state change of one site, and both reactant species remain unbound. Hence, it is a simplification of Michaelis-Menten, in which reversible binding can trigger a chemical reaction and rapid dissociation, here assuming the chemical reaction is instantaneous; e.g., A(a) + B(b∼u) → A(a) + B(b∼p).

Coupling reactions

NERDSS allows the completion of one reaction (such as dissociation of a complex) to be coupled to another reaction occurring immediately. Thus, one can readily model enzymatic reactions and Michaelis-Menten kinetics. In particular, if an enzyme binds a substrate, the dissociation reaction can be directly coupled to a state change on the substrate, for example, from unphosphorylated to phosphorylated.

Intracomplex binding and defect formation

An important consequence of forming self-assemblies is that molecules can be capable of binding to one another through free sites when they are within the same complex. These intracomplex binding events involve two sites becoming a bound complex but are not truly bimolecular because they do not involve a search to find one another—they are already colocalized. They are thus comparable to conformational rearrangements and are dependent on time (and fundamentally on the details of the atomic interactions) and not on the system size or density. This type of unimolecular or first-order reaction cannot be treated with the same binding probabilities as are used for bimolecular events despite it involving a bond forming between two species. In particular, for the rigid species that we model here, they will have no motion relative to one another (D_tot = 0), and as noted previously (44), this results in binding probabilities of either one or zero, with the former violating detailed balance and the latter preventing the formation of important stabilizing bonds. Even with flexible molecules, in which intracomplex sites can undergo relative diffusion, constraints on mobility due to the bonded structure would, over most time steps, require adjusted reaction probabilities. We define the binding probability of these intracomplex or loop-closure events thus using a Poisson probability, similar to the first-order reactions described above. Then, we must specify a unimolecular rate, given a bimolecular rate constant. We define this rate such that the equilibrium between the bound and unbound states for a two-step process of loop closure, that is, a bimolecular event and a unimolecular loop closure, is the same as if the protein closed the loop in a single step, forming both bonds at once. The rate is given by

$$k_{close}=k_a{C}_0\text{exp}\left(-\Delta G_{coop}/k_{B}T\right),$$

where C₀ is the standard state concentration of 1 M. For loop closure, the user can thus specify one additional parameter, ΔG_coop. By default, ΔG_coop = 0, and loop-closure events are typically difficult to reverse, requiring two bond-breaking events to separate the loop back into two pieces. Positive values of this parameter render loops less stable and more reversible or dynamic (Fig. S3). We derive this expression in the Supporting Materials and Methods.

We allow these events to form when the sites are spatially proximate, with the user able to specify a maximum distance to allow these events to occur. The maximum distance is the binding radius, σ, multiplied by a factor, bindRadSameCom, which by default is set to 1.1, which includes only perfectly (with numerical precision errors allowed) aligned contacts. The primary reason to increase it is because for some self-assembly structures (such as the curved clathrin cage), the rigid structures cannot form perfect, defect-less structures (see Fig. S8). Allowing binding between legs that are close together mimics structural flexibility present in the biological molecules. A more sophisticated treatment in the future could couple defects to energetic penalties that impact reaction rates and allow relaxation within the rigid assembly as deviations relative to the user-defined orientation angles. Finally, the software prohibits binding between a pair of molecules if they are already bound through a distinct set of sites. Hence, intracomplex or loop-closure events must be mediated through at least a third molecule (e.g., four molecules in the case of clathrin lattices).

Evaluation of steric overlap

Once association events occur between two sites, their two parent complexes are rotated into place to generate the proper, predefined geometry of the two binding proteins. The exception is intracomplex binding, in which because the binding happens within a single complex, no rotation or movement occurs. For two separate complexes, however, this rotation into place can result in steric overlap between sites that are part of the complex but not part of the binding event. We evaluate overlap after association by measuring distances between all centers of mass (COMs) in the new complex and defining a minimum distance, overlapSepLimit, that will determine steric overlap. If any pair of proteins have COMs less than overlapSepLimit, which by default is 1 nm, the binding event is rejected, and the two complexes instead undergo a diffusion move during the time step. This is the simplest evaluation of steric overlap and could be improved, particularly in cases of defect-forming structures (Fig. S9). Here again, more elaborate schemes could determine whether the full space occupied by a molecule overlaps with other molecules, not just the COM.

A final steric overlap check is then performed between the newly bound complex and all other binding sites of complexes present in the simulation to ensure excluded volume is maintained. If the new complex produces steric clashes with other binding sites in the simulation volume, the binding event is rejected.

Excluded volume

Sites that react with one another have excluded volume that is determined by their binding radius, σ. Specifically, in 3D, a reactive site excludes all partners from entering within a sphere of radius σ, and in 2D, within a disk of radius σ. By default, sites otherwise do not exclude volume with each other but act as point particles, including the COM sites. Therefore, if one desires sites to exclude one another, they must be specified as undergoing a bimolecular reaction, and by setting the rate to zero, they will reflect off one another at σ without ever binding. We note that the addition of reactions inevitably slows down the simulations.

An important consequence of having excluded volume only between specified reactive pairs is that once a pair of sites is in the bound state, they no longer automatically exclude volume from other sites in the system. For proteins that are fully bound (and thus are temporarily nonreactive to bind), they can thus pass through other proteins, which is often physically undesirable. To maintain excluded volume, proteins can be given dummy sites that are present purely to exclude volume with dummy sites on other molecules, at all times. A future alternative will be to have an option for all bimolecular reactions that sites can continue reflecting off their reaction partners, whether they are in the free or bound states.

Boundary effects

By default, the simulation boundaries are reflecting. Rectangular and spherical geometries are implemented, with no restriction on user-defined size. If association events result in a very large lattice, it is possible for the structure to extend beyond the physical boundaries of the simulation volume. These moves are thus rejected. On a related note, if association events result in very large rotational reorientation of a complex, these moves can also be rejected because they result in an unphysical amount of displacement per time step. Binding to lipids or sites restricted to the surface can be defined using explicit lipids or a field-based implicit lipid model (74).

Diffusion constants of complexes

Once proteins bind to form a new complex, their translational and rotational diffusion constants are updated to reflect the larger hydrodynamic radius of the bound complex. Each protein molecule has a user-defined D_t and D_R. Once a bound complex forms, the new transport coefficients are then defined based on all N components of the complex by simply assuming the radii sum:

$$D_{\mathrm{t},\mathrm{x}}={\left[\sum _{i=1}^N{D}_{\mathrm{i},\mathrm{x}}^{-1}\right]}^{-1},$$

$$D_{\mathrm{R},\mathrm{x}}={\left[\sum _{i=1}^N{D}_{\mathrm{R}\mathrm{i},\mathrm{x}}^{-1/3}\right]}^{-3},$$

and the same in y and z. For molecules restricted to the surface, D_t,z = 0 and all rotational diffusion except (if desired) D_R,z are also zero. Correspondingly, any complex that contains these molecules will also have that diffusional component set at zero.

Data output

NERDSS produces restart files that ensure that when simulations are interrupted for any reason, they can be restarted exactly from where they left off, similar to molecular dynamics software. NERDSS also allows restarts to introduce new molecules or reactions into the system using input files formatted the same as the standard inputs. This allows systems to be prepared or equilibrated in one state before, for example, enzymes are introduced.

Simulation algorithm outline

• 1)Initialization. Copies of rigid species are created from the provided molecule templates, given random coordinates, and checked for placement within the simulation volume and overlap with other particles. If two reactive partners are found to be overlapping, one of the particles is reinitialized with new random coordinates, and all particles are rechecked for overlap.
• 2)Optimization. To optimize evaluation of the two-body (bimolecular) events, the simulation volume is split into N sub-boxes, in which the length of each sub-box edge is at least R_max,sys = $\underset{m}{\max }\left\{{\sigma }_m+3\sqrt{6D_m\Delta t}+l_{1m}+l_{2m}\right\}$, where m loops over all binding reactions, σ is the binding radius, D the total diffusion coefficient of both reactants, and l_im is the radius of the reactant protein i. Molecules are assigned to boxes based on the position of their COM. This allows for looping over all possible binding partners only in a molecule’s current and neighboring boxes, without excluding any partners with a nonzero reaction probability.
• 3)Reactions. Within each time step, reaction probabilities are calculated in order from zeroth-, first-, and second-order reactions. Reactions are described in detail above. We note that if a molecule is created or involved in a reaction in any way, it cannot participate in any other reaction during the course of the time step, nor can it diffuse. Hence, only one site per molecule can react per step. Other diffusing molecules will continue to include these molecules in overlap checks to prevent overlap in subsequent steps.
• 4)Propagation and overlap. After all reaction checks are completed, sites that were within R_max of a reaction-partner during the time step are checked for overlap. Because a multicomponent complex only moves as a rigid unit, each complex with reactive sites within is looped over, along with a stored list of all reaction partners. If the newly displaced positions overlap (with displacements sampled from Gaussians for translational and rotational diffusion for both reaction partners), the displacements of complex i and its partner are resampled, and the overlap check restarts. Once a complex’s position has been updated, it cannot be moved again.

Simulation details and parameters

Simulation parameters are listed in Table 1, and values are included in figure legends. Transport coefficients are defined using the Einstein-Stokes equation, using each molecule’s hydrodynamic radius estimated from its approximate radius of gyration. Unless otherwise noted, σ = 1 nm for binding.

Table 1.

Parameters for NERDSS Simulations

Name	Unit	Meaning
Δt	μs	time step
kon	μM−1s−1	macroscopic reaction rate
koff	s−1	macroscopic unbinding rate
ka	nm3/μs	microscopic/intrinsic reaction rate
kb	s−1	microscopic/intrinsic unbinding rate
KD	μM	dissociation constant
Dt	nm2/μs	translational diffusion coefficient/constant
DR	rad2/μs	rotational diffusion coefficient/constant
Dtot	μnm2/μs	net diffusion coefficient of one reactant relative to another
σ	nm	binding radius
Rmax	nm	largest distance within which binding reactions can occur
ΔGcoop	kBT	adjustment of the free energy during the loop-closure event
f		loop cooperativity factor, f = exp(−ΔGcoop/kBT)
bindRadSameCom		Multiplier σ, sets max distance between sites to allow the loop-closure event to occur
overlapSepLimit	nm	Max distance between molecule COMs resulting in steric overlap

We describe here the geometry of the molecules from the clathrin and Gag systems and the orientation angles.

Clathrin model

For the flat clathrin molecule, each clathrin trimer has a COM and three leg binding sites rotated 120° relative to one another, and each leg site is 10 nm from the center such that when dimerized, the centers are 21 nm apart, which is comparable to experimental lattices (75). When two clathrin molecules bind to each other, we suppose their leg sites are head to head and on a line with their centers, and, to generate a flat hexagon structure, all their centers and legs should be on the same plane. Thus, the five orientation angles are [θ₁, θ₂, φ₁, φ₂, ω] = [π, π, nan, nan, 0] (“nan” means no value constraint, and see Fig. 1 and Supporting Materials and Methods for the definition of orientation angles). For binding to adaptor proteins, each flat clathrin includes three more leg sites, which we choose to locate directly beneath the legs (π/4.656 angle between the long leg and short leg), as shown in Fig. 5 a. When the clathrin binds to a linear adaptor protein, the orientation should make the long legs parallel to the membrane, where the adaptor and lipid site are designed to be perpendicular to membrane surface. Therefore, the five orientation angles of adaptor to clathrin (adaptor is molecule 1) should be set as [π/2, 3.656π/4.656, nan, π, π]. For the puckered clathrin, the legs are tilted below the plane (5, 10, or 20°) and rotated 120° around the molecule normal, with l_leg = 7.5 nm. The orientation angles of two bound puckered clathrin is the same as flat clathrin [π, π, nan, nan, 0]. The adaptor binding to the lipid is two linear molecules binding with angles [π/2, π/2, nan, nan, π].

Figure 5.

Clathrin assembly and disassembly by membrane localization and delocalization. (a) Three multivalent solution proteins are included in this model (clathrin, adaptor protein, phosphatase) along with one membrane lipid (PI(4,5)P₂). (b) Clathrin contains three sites for binding other clathrin and three separate sites for binding to the adaptor. The adaptor protein has a dedicated site for each of the three other components, which is typical of a wide range of clathrin adaptors (83). This allows clustering on the membrane. The PI(4,5)P₂ is present comparable to its physiologic value, ∼1 mol % of plasma membrane lipids (92). The phosphatase has independent sites for adaptor and lipid binding. (c) With no enzymatic activity, the assembly proceeds downhill, directly to an equilibrium steady state (red curves). With phosphatase activity turned on, the majority of the lipids are converted by 1 s, except for the population that has been already bound to adaptors. The clathrin lattice thus has time to assemble but is gradually destabilized as determined by the rate of unbinding between adaptor and lipid and adaptor and clathrin, which increases from 1 to 10 s⁻¹ between cyan and blue. N_cla = 100 clathrin trimers, N_adaptor = 300, N_pip2 = 6000, N_phos = 10. Rectangular box of size [0.7, 0.7, 0.494] μm, flat and immobile membrane on the bottom. Δt = 0.2 μs. k_off^Cla-Cla = 10 s⁻¹. k_on^Cla-adap = 6 μM⁻¹s⁻¹ k_off^Cla-adap = 1 s⁻¹, 10 s⁻¹. k_on^pip2-adap = 6 μM⁻¹s⁻¹, k_off^pip2-adap = 1 s⁻¹, 10 s⁻¹. k_on^phos-adap = 0.6 μM⁻¹s⁻¹, k_off^phos-adap = 1 s⁻¹, k_on^pip2-phos = 25 μM⁻¹s⁻¹, or 0 for “No activity.” For all 2D binding interactions, k_a^2D = k_a^3D/2σ, k_b is unchanged, σ = 1 nm, f = 0.001, D_cla = 13 μm²/s, D_R,cla = 0.03 rad²/s. D_ap = 25 μm²/s, D_R,ap = 0.5 rad²/s. D_pip = 1 μm²/s (D_z,pip = 0), D_R,pip = 0. D_syn = 25 μm²/s, D_R,syn = 0.5 rad²/s. To see this figure in color, go online.

Gag model

The Gag lattice will form a sphere of radius ∼50 nm because we design the diameter of each hexamer to be ∼8 nm and the angle between two hexamers is ∼9.2°, comparable to experimental lattices (76). Relative to the COM, the Gag has homodimerization sites at [0, 0.5 nm, 0]. The homodimerization angles are [θ₁, θ₂, φ₁, φ₂, ω] = [1.49, 1.49, 0, 0, 0]. The heterodimerization sites are set at [−2 nm, 0, 0] and [2 nm, 0, 0], such that the side of each hexamer is ∼4 nm (slightly longer because of the binding radius). The five angles are [2.62, 2.62, π, π, 0]. Our Gag monomer has been designed with binding sites for membrane lipids and RNA binding sites, so future work can effectively probe assembly kinetics as controlled by Gag binding partners and membrane localization.

Results

NERDSS enables self-assembly simulations

NERDSS expands beyond existing single-particle RD software by propagating rigid, user-defined, orientable components that assemble rigid multicomponent structures such as hexagonal clathrin lattices (Fig. 1 a). The algorithmic elements of NERDSS that are new for single-particle and spatial RD (Fig. 1 b) arise because of this propagation of rigid multisite molecules (44), which requires generalized routines to orient molecules and complexes once they bind, corresponding assessment of steric overlap and defect contacts, integration of rule-based control of each site, and generalized output for visualization and analysis (see Methods). We provide a GUI to facilitate the otherwise cumbersome construction of rigid molecules and bound states. The binding between molecules is independent of their orientation before the event (44), which is true of all single-particle methods, including those with interaction potentials, because of their spherical symmetry (14,40). NERDSS is efficient enough to simulate seconds to minutes of assembly on a single processer (Fig. S1), primarily because of algorithms that support relatively large time steps (32). Localization and assembly on membranes also can proceed efficiently and accurately by using a recent algorithm for implicit lipids, which avoids the need to propagate large populations of individual lipid binding sites (74). Although not yet optimized for parallel or GPU architectures (27,34,40), this is an active development. The significant advantages of NERDSS are the types of biological model systems it makes accessible and the accuracy of the single-particle algorithms used for propagating species, illustrated in the next section.

Validation of NERDSS kinetics and equilibria

We validate the NERDSS software first for several non-self-assembly models to verify fundamental reactions and the rule-based implementation of the RD model either against theory or other simulation methods. Zeroth- and first-order reactions reproduce the exact analytical solutions as expected, and they are not dependent on space (newly created particles are placed randomly in the volume) (Fig. 2 a). Bimolecular reactions for single-particles have been previously validated for the FPR algorithms in 3D (32), 2D (73), and transitions from 3D to 2D with flat and curved surfaces (74), but we perform additional verification here (Fig. 2 b). The FPR algorithms are derived to preserve detailed balance for reversible binding interactions, ensuring thermodynamic equilibrium is reached when systems are designed to (excluding kinetic traps). Binding to the surface can be mediated through explicit lipid sites, or we have implemented an implicit lipid model that is substantially more accurate than surface adsorption models at reproducing binding kinetics of the explicit lipid simulations (74). This model provides orders of magnitude speedups relative to explicit lipid simulations (74) and couples with all other reaction types (Fig. 2 c). Although NERDSS is compatible with arbitrarily curved surfaces (77), the user inputs currently accommodate only spherical geometries (Fig. 2 c) or rectangular volumes in which the flat membrane is on one surface. We propagate diffusion on the sphere by exactly sampling displacements on the spherical surface (Supporting Materials and Methods), and all distances between surface-bound particles are measured using geodesics.

Figure 2.

Multiple validations of NERDSS algorithms and features. (a) Creation and destruction are compared with analytical theory from the corresponding oridinary differential equation (ODE), where k_C = 0.01 M/s and k_d = 1000 s⁻¹, V = 1 μm³. (b) Bimolecular reactions in different dimensions are shown. 3D and 2D analytical theory curves from the ODE, 3D → 2D theory as the numerical solution to the PDE are shown, solved using Virtual Cell. The kinetics of the 2D single-particle RD solution is theoretically known to differ from the continuum solution (73). For all reactions, σ = 2 nm. For 3D: V = 0.83 μm³, D_A = D_B = 20 μm²/s, k_on = 300 μM⁻¹s⁻¹, k_off = 50 s⁻¹, A₀ = B₀ = 2 μM. For 2D: A = 1 μm², D_A = D_B = 2 μm²/s, k_on = 10 μm²s⁻¹, k_off = 50 s⁻¹, (k_a^2D = 100 μm²s⁻¹), A₀ = B₀ = 800/μm². For 3D → 2D: cube of L = 1 μm. D_A = 40 μm²/s, D_B = 2 μm²/s, k_on = 100 μM⁻¹s⁻¹, k_off = 50 s⁻¹, A₀ = 1.412 μM, B₀ = 3000 μm². (c) Localization and binding on the membrane (as modeled in (45)) with a box geometry and a sphere geometry produce the same equilibrium and kinetics. This model combines all bimolecular reactions tested in part (b) and includes diffusion on the curved surface and the implicit lipid model (see Fig. S5 for parameters). (d) Clathrin trimer self-assembly, where theory is the analytical solution to the ODE for reversible binding between identical, independent sites, is shown. V = 0.494 μm³, N_trimer,₀ = 100, k_on = 1 μM⁻¹s⁻¹, k_off = 1 s⁻¹, D_trimer = 13 μm²/s, D_R = 0.03 rad²/s. <N_CC>_eq = 104.7 and 104.8 for Δt = 0.1, 0.2 μs respectively. The NERDSS simulations capture the spatial lattice formation, and thus, their equilibrium values also depend on the formation of closed loops, where here f = 5.9e−6, bindRadSameCom = 5 nm, and overlapSepLimit = 7 nm. To see this figure in color, go online.

To validate the self-assembly simulations, we follow previous work (44) by comparing our results against nonspatial rate-based simulations and by comparing our simulation results as we change the time step of integration (Fig. 2 d). Changing the time step is a valuable test for two reasons: first, as the time step decreases, the assumption that each multisite molecule is interacting with only one other site becomes more and more accurate, which improves adherence to solving only two-body problems per step (32). Second, the reaction probabilities depend nonlinearly on the time step, so if inaccuracies existed in diffusion or reactions, they show up quantifiably in the kinetics and equilibria. We performed the same validation for assembly in 2D (Fig. S4). For self-assembly, we must introduce a new loop-closure reaction type (Methods), and we show how the choice of this parameter, which controls the binding free energy of loop-closure events, impacts the equilibrium (Fig. S3). We cannot compare NERDSS self-assembly with other single-particle software because they do not capture orientation-dependent interactions and thus cannot form a structured lattice or cage. It is informative to compare NERDSS to NFSim (8), which lacks spatial resolution but simulates rate-based models that can form multiprotein assemblies via multisite species and rule-based modeling (Figs. S2 and S3). In weak-binding regimes, they agree exactly, but with stronger binding, the structure resolution in NERDSS drives spatially localized interactions and captures several elements of self-assembly not possible with NFSim, including lattice geometries and topologies and their growth patterns.

To test the rule-based implementation with a more complex nonassembly model, we simulated a previously developed minimal model of a circadian genetic oscillator (53) with NERDSS, as well as with PDEs and the stochastic simulation algorithm (via Virtual Cell software (6)), showing quantitative agreement across all models (Fig. 3). Our simulation recapitulates how the expression of two proteins, an activator protein (A) and repressor protein (R), can be coupled to produce robust oscillations of both proteins and their bound complex. For our NERDSS model, when messenger RNA and proteins are created, they are placed adjacent to the molecule creating them. All rates were accelerated by a factor of 3600 relative to the original rates because of computational costs, such that oscillations occurred over timescales of seconds rather than hours. This change made the binding events diffusion limited, meaning they could become sensitive to the spatial distribution of particles. However, because the species mix rapidly and the volume is not too large (Video S5), the kinetics were essentially insensitive to diffusion and the spatial dimension, indicating the resolution offered by NERDSS is not necessary for this model. We note the PDE and nonspatial simulations have significantly faster CPU time.

Figure 3.

NERDSS simulations of a circadian clock model recapitulate oscillatory protein expression. (a) The model contains six components, with the activator (A, in red) and repressor (R, in blue) proteins produced by their corresponding messenger RNA, which are produced by a single copy of each gene (PmrA and PrmR). All species have D_t = 10 μm²/s, and initial copies were zero except for one copy of PrmA and PrmR. Unimolecular reactions (except dissociation) are shown with reactions indicated by η or δ (η₁ = 50 s⁻¹, η₂ = 0.01 s⁻¹, η₃ = 50 s⁻¹, η₄ = 5 s⁻¹, η₅ = 500 s⁻⁻¹, η₆ = 50 s⁻¹; δ₁ = 1 s⁻¹, δ₂ = 0.2 s⁻¹, δ₃ = 10 s⁻¹, δ₄ = 0.5 s⁻¹) binding and unbinding reactions are shown with rates π (π_a1 = 1204 μM⁻¹s⁻¹, π_a2 = 602 μM⁻¹s⁻¹, π_a3 = 602 μM⁻¹s⁻¹, π_b2 = 100 s⁻¹, π_b3 = 50 s⁻¹, u_b = 1 s⁻¹). For bimolecular association reactions, NERDSS converts the macroscopic rates above to microscopic rates, where σ = 5 nm for binding PrmA/R and σ = 8 nm for A + R. One exception is unbinding of the complex A-R (magenta dashed). Because unbinding results in only R (A is modeled as degraded instantaneously), there is no possibility for geminate recombination, and thus, the unbinding follows the macroscopic rate u_b. (b) Solution using a PDE solver produces regular oscillations, where the calculated oscillation time is 25.1 s. (c) The oscillation time of A (and R) from NERDSS is in close agreement with the PDE-based simulations, calculated as 24.5 s. In the Supporting Materials and Methods, we show similar results for a nonspatial model (time = 24.8 s). Simulation results were the same with Δt = 10 or 50 μs. Because of the constant production and degradation of components in this model, we used Ovito software to produce videos for this model (Video S5) and the Gag model (Video S4). To see this figure in color, go online.

NERDSS tests clathrin cage assembly designed by four distinct models of clathrin-clathrin interactions

Clathrin, a 600 kDa trimeric protein, assembles into both flat and spherical lattices in vivo and in vitro (1,78,79), although in vivo this process only occurs on the plasma membrane, with the requirement of a host of accessory and adaptor proteins, because clathrin itself does not bind the membrane. Clathrin trimers are observed to assemble in solution in vitro either at low pH or in the presence of clathrin terminal-binding adaptor proteins (80,81). We model distinct assembly conditions by changing binding strengths, cooperativity, and binding geometries of clathrin-only solutions, contrasting how multiple observables vary with distinct model parameterizations at a fixed (1.3 μM) clathrin concentration. The baseline model has a K_D for leg-leg binding of 100 μM, which is comparable to experimental estimates of trimer dimerization (82). Very little binding is observed in solution at this concentration (Fig. 4 a), which is consistent with in vitro studies of clathrin in solution at physiological pH (81). The same results are observed in nonspatial simulations (Fig. S3).

Figure 4.

NERDSS simulations of designed clathrin lattice assemblies in solution, with 1.3 μM clathrin triskelia. (a) For K_D = 100 μM, most clathrin remains monomeric in solution. The second column shows corresponding counts of pairs of bound clathrin trimer legs in time. The third column shows a histogram of the total copies of clathrin triskelia as distributed complexes of increasing size. (b) Strengthening K_D to 0.2 μM is shown. (c) The cooperative model has binding of monomeric to nonmonomeric clathrin stronger by a factor of 10, and nonmonomeric to nonmonomeric is again 10 times stronger. Lattices can now form despite a weak monomeric K_D = 100 μM. (d) By changing geometry of the clathrin monomers (K_D = 0.2 μM), they now assemble into spherical cages with similar binding kinetics but that nucleate smaller complexes (d₃ relative to b₃). Results are averaged over three to five trajectories, and Supporting Videos are generated with VMD (90). Cube with L = 0.494 μm, k_off = 1 s⁻¹, and f = 0.001. N_trimer,₀ = 100, D_trimer = 13 μm²/s, D_R = 0.03 rad²/s, Δt = 0.2 μs. Flat clathrin molecules: leg length l_leg = 10 nm. Puckered clathrin, l_leg = 7.5 nm, and bindRadSameCom = 5 nm, overlapSepLimit = 7 nm. To see this figure in color, go online.

By simply strengthening the K_D, the flat clathrin lattices start to assemble rapidly, to the point that a K_D = 1 μM produces a similar equilibrium in bound legs as 0.2 μM (Figs. 2 d, 4 b, and S10). As we strengthen K_D, average size of complexes transitions to a giant component at ∼10 to 1 μM. With the formation of closed loops in the clathrin lattice, we can introduce positive or negative cooperativity, due to the formation of two simultaneous bound interfaces within the closed loop (Methods and Supporting Materials and Methods). This can thus alter the final equilibrium for a given dimerization K_D, favoring fewer bonds and more remodeling (Figs. S3 and S10), and will also shift the value of k_on when nucleation of complexes progresses to complete lattices.

By instead introducing cooperativity in binding between clathrins such that the rate depends on whether a monomer is already bound to another clathrin, flat lattices assemble after a delay, followed by rapid growth (Fig. 4 c). This model thus has three on-rates for clathrin-clathrin binding, with strongest binding between clathrins that are already bound to others. This growth model results in a broader distribution of lattice sizes (Fig. 4 c; Video S1).

Video S1

Clathrin Lattice in Solution

By taking the model of Fig. 4 b (K_D = 0.2 μM) and altering the geometry of a clathrin monomer to tilt legs 10° below the plane (100° pucker), spherical cages form (Video S2). The kinetics of leg-leg binding remains very similar to Fig. 4 b, but the sizes of the complexes are smaller due to spatial constraints of closed spheres and steric overlap (Fig. 4 d). The extent of the tilt controls the radius of the sphere, with this 100° clathrin producing a sphere of R ∼45 nm containing ∼100 clathrin. A 95° clathrin, in contrast, produce a sphere of R ∼92 nm, estimated to contain ∼500 trimers, and a 110° degree clathrin produces a sphere of R ∼25 containing ∼20 trimers (Fig. S7). We note that the rigid clathrin trimer cannot form a perfect spherical lattice, and thus, defects appear even within a single closed hexagon. In the most puckered clathrin (110°), the geometry much better accommodates pentagons, not hexagons (Fig. S7). Biomolecules typically have the flexibility to bend and form contacts by supporting a distribution of structural geometries, and clathrin is known to form cages in solution and on the membrane at a range of curvatures (75). We thus allow sites within a complex to bind one another even when they are not at perfect contact (Methods). The choice of cutoff distances for binding within a complex and for identifying steric overlap (which is more difficult with defects) does impact the stability of the cages and whether they prevent unphysical overlap (Figs. S8 and S9); more sophisticated treatments in the future will improve the treatment of defect formation (Methods).

Video S2

Clathrin Cage in Solution

NERDSS probes lattice assembly driven by localization to the membrane

Increasing component concentrations is another natural mechanism to nucleate assembly. In Fig. 5, we show how localization to the 2D membrane surface can, on its own, nucleate clathrin lattices by increasing the effective concentration of clathrin relative to the 3D solution (Video S3). Clathrin does not bind the membrane directly but can be localized to the surface by an adaptor protein. The clathrin-clathrin interaction is set here to a weak value of 100 μM (Fig. 5 a; (82)), and thus, minimal binding occurs in solution. Once the clathrin have been localized to the surface, however, they can assemble into a lattice that is stabilized by relatively weak clathrin-clathrin contacts that are favorable because of the small search space available on the 2D surface (45). Although binding to many adaptor proteins is known to stabilize clathrin-clathrin interactions (80), we do not incorporate this cooperativity here to isolate the role of 2D localization in nucleating assembly.

Video S3

Clathrin Lattice on Membrane

Enzymes can drive lattice disassembly by removing links to the membrane

By removing links between clathrin and the membrane, we can drive the clathrin assembly back into solution where our clathrin lattice is no longer stable. Physiologically, this is achievable by changing the phosphorylation state of the lipid PI(4,5)P₂, which is essential for localizing adaptor proteins, and thus clathrin, to the membrane (83). Thus, without altering the clathrin or the adaptor proteins directly, we can drive disassembly of the lattice. We include here 10 copies of a lipid phosphatase (e.g., synaptojanin), which converts PI(4,5)P₂ to PI(4)P irreversibly. Critically, the adaptor protein cannot bind to PI(4)P, thus removing its link to the surface. Importantly, the phosphatase synaptojanin can be localized to sites of clathrin-coated structures in vivo through protein interactions with the adaptor protein AP-2⁸⁵, thus allowing it to act in 2D to rapidly dephosphorylate unbound PI(4,5)P₂. Although in vivo, it has been shown to play an important role in dephosphorylating PI(4,5)P₂ after fission from the membrane (59), we demonstrate here how it is capable of driving disassembly of plasma-membrane-bound clathrin-coated lattices, at least in this in vitro-type model.

We find the timescales of the lattice disassembly are sensitive to the binding and unbinding kinetics of all protein-protein and protein-lipid interactions (Fig. 5). In particular, although the phosphatases rapidly dephosphorylate unbound PI(4,5)P₂, the adaptor proteins protect the subset of lipids they are bound to, and their links are cut on a timescale that is determined by unbinding of the adaptor-lipid interactions. Because clathrin can bind up to three adaptor proteins, its dissociation from the membrane then also depends on dissociation of all three adaptors from the surface. Although this model is not fully parameterized to reproduce experiment, it illustrates how localization, enzymatic activity, and binding kinetics can be tuned to control assembly and disassembly. Equilibrium constants used are within reasonable ranges of experimental measurements for endocytic proteins (collected here (45)).

Model building requires orientations for each pairwise binding reaction: application to viral assembly in solution

To illustrate the model design process, we develop a coarse model of the Gag monomer, a retroviral protein that dimerizes with itself and oligomerizes into a hexagonal lattice on the plasma membrane of infected cells to form the so-called immature virion (72). To form a lattice of hexamers from these approximately linear monomers, two distinct types of dimers are sufficient, a homodimer and heterodimer interaction, with relative orientations defined by five separate angles for each dimer (Fig. 6). The heterodimer forms between head-to-tail monomers, allowing polymer-like growth at a specific hexameric geometry. The lattice forms with sizes of hexameric subunits (∼8 nm in diameter) and a spherical curvature (R ∼50 nm) that would support an inclusion of ∼2500 Gags in the full spherical shell, as observed experimentally (76). The Gag-Gag homodimerization is stronger than other Gag-Gag contacts at an estimated 1–10 μM and appears to form first (84). The weaker Gag-Gag hexameric interactions are known to be autoinhibited before binding to either RNA, negatively charged analogs, or the membrane (72,84); however, here we treat all interactions as constitutive for simplicity. Our Gag monomer has been designed with binding sites for membrane lipids and RNA binding sites, so future work can effectively probe assembly kinetics as controlled by Gag binding partners and membrane localization.

Figure 6.

Gag monomer assembly model setup and titration experiment. (a) Each Gag protein monomer is given three active binding sites and two inactive binding sites, positioned relative to a COM site in red. The blue site is for homodimerization, H, and the green (G1) and orange (G2) sites are for heterodimerization to produce hexamers. The five orientation angles are based on the COM to binding site vectors (turquoise lines), the sigma vector (purple line), and two molecule normals (black vectors) that must not be co-linear with the site vectors to provide an additional dimension (Methods). The orientation of the bound heterodimer similarly requires five angle definitions, and G1 must bind G2 at a cis orientation to ensure a hexamer loop can form (inset). (b) Trajectories show the titration in of Gag monomers at 3.3 × 10⁻⁵ and 6.6 × 10⁻⁵ M/s. (c) The number of monomers initially grows rapidly, but reaches a relative steady state because of assembly and degradation, which occurs at a rate of 1 s⁻¹. Assembled proteins are protected from degradation, causing a continual growth in sizes of assembled complexes. Box = [1, 1, 1] μm. Δt = 0.1 μs, k_on^H-H = 3 μM⁻¹s⁻¹, k_off^H-H = 1 s⁻¹. k_on^G1-G2 = 0.3 μM⁻¹s⁻¹, k_off^G1-G2 = 1 s⁻¹, D_t = 10 μm²/s, D_R = 0.05 rad²/s. f = 0.001, bindRadSameCom = 1.5 nm, overlapSeplimit = 0.7 nm. k_deg = 1 s⁻¹ (only fully monomeric Gag). To see this figure in color, go online.

To illustrate the functionality of NERDSS, we initialize the volume with zero Gag monomers. We then titrate them in using a zeroth-order reaction, for which they are placed randomly in the simulation volume. Thus, the concentration slowly increases, favoring slower nucleation and more efficient growth of the Gag into a small number of larger assemblies (Video S4). Monomeric Gag is also degraded, although assembled Gag is not. Thus, the long-time behavior consists of Gag as stable spheroids or as short-lived (∼1 s) monomers. As designed, the Gag is able to assemble into large spheroids entirely because of the two specified reactions (Fig. 6). The assembly kinetics depends on the titration speed, which can provide additional knobs to help optimize the model against experimental data.

Video S4

Gag Cage in Solution

Video S5

Circadian Clock Oscillator

Discussion

NERDSS has several advantageous features that make it a powerful and immediately useful tool for cell-scale simulations. First, NERDSS is transferrable between distinct systems thanks to its rate-based interaction framework, which avoids the time-consuming parameterization of energy functions often hard coded for specific assembly systems (19). Although protein geometries and orientations of bound complexes in NERDSS are system specific, we provide a GUI to facilitate user design of proteins and their bound states. Second, unlike existing spatial rate-based approaches (33), the built-in molecular structure of NERDSS not only enforces excluded volume of binding sites but evaluates whether steric overlap or volume restrictions would prevent the formation of unphysical assembly structures. Third, it uses the FPR algorithms (32,44,73) to efficiently propagate species while retaining accurate rates of association and dissociation, allowing for current simulations of timescales on the order of seconds to minutes. NERDSS converts carefully between microscopic rates used in FPR and macroscopic rates commonly defined from experiment, with extensive validation (44). Fourth, NERDSS uses a BioNetGen Language-style syntax (46), and models built in other software packages using similar syntax can be ported to NERDSS with minimal alterations, making it available for immediate use. In this combination of structural resolution, efficient propagation and accurate treatment of association rates, and user-friendly input file syntax, NERDSS is a unique tool for simulating equilibrium and nonequilibrium self-assembly and other cell-scale phenomena.

A current limitation of NERDSS is that all species must be rigid bodies, and thus, assembly defects that arise within complexes that cannot perfectly tile surfaces (such as rigid bodies on a sphere) are accommodated approximately by using a distance tolerance. Molecule flexibility (14) is a natural extension beyond rigid molecules that would improve defects and also enable better treatment of disordered regions and genomes. NERDSS also lacks any orientation dependence on binding between complexes before association, which is an approximation that can result in unphysical binding events, particularly between large multicomponent intermediates, as also discussed previously (44). Although we have implemented options to reject these moves, introducing orientation dependence to reaction probabilities (13,85) would be more physically realistic and could allow NERDSS to predict a range of structures rather than the ones predetermined by the defined geometries, although this would require more expense or approximate. NERDSS is designed to be extensible to not only users but to developers expanding functionality. The FPR algorithms support, for example, the implementation of more elaborate physical models that introduce electrostatic interactions between particles (13,32). A particularly exciting future development is integration of NERDSS with continuum membrane models (74) to allow more realistic simulations of vesicle and virion formation dynamics and coupling of assembly to mechanical force generation, which is a key component, for example, in cytoskeletal assembly models (86) and continuum membrane remodeling studies (87, 88, 89).

NERDSS is designed to be more user friendly and portable than previous versions of the FPR algorithms (44) with the addition of rule-based modeling, the formatting of input and output files, and the addition of the GUI, all available open source here: github.com/mjohn218/NERDSS. However, it does not match the usability of software such as Virtual Cell in terms of model setup and analysis, including defining cell geometries from experimental data (6,52). It also does not match the level of parallelization or optimization of software such as Lattice Microbes (27). These will be important future software developments to support a broad and diverse user base.

We have introduced self-assembly models here motivated by CME and virion formation in cells as a foundation to extend these models to additional components and experimental comparison. We show how assembly can be nucleated despite weak clathrin-clathrin interactions through cooperativity or through reduction of dimensionality to the membrane (3D to 2D) (45). With NERDSS, we predicted here how the dephosphorylation of essential plasma membrane lipids, which occurs at sites of vesicle budding (59), could de-stabilize clathrin-coated structures and drive their disassembly. Ultimately, NERDSS has encoded as a core feature the ability to resolve relatively fast processes over long timescales and individual proteins over long length scales, simulating otherwise intractable protein assembly dynamics. This provides immediate use in helping overcome the challenges of understanding or designing self-assembling structures in biology.

Author Contributions

M.J.V., Y.F., and M.E.J. wrote the software. Y.F. implemented the implicit lipid and spherical model to the software. O.N.Y. and S.L. designed and developed the GUI. M.J.V., Y.F., and M.E.J. ran simulations, collected results, and wrote the article.

Editor: Padmini Rangamani.

Supporting Material

Document S1

Supporting Materials and Methods, Figs. S1–S10, and Table S1