Computer simulations of lipid regulation by molecular semigrand canonical ensembles

Abstract

The plasma membrane is the interface between cells and exterior media. Although its existence has been known for a long time, organization of its constituent lipids remain a challenge. Recently, we have proposed that lipid populations may be controlled by chemical potentials of different lipid species, resulting in semigrand canonical thermodynamic ensembles. However, the currently available molecular dynamics software packages do not facilitate the control of chemical potentials at the molecular level. Here, we propose a variation of existing algorithms that efficiently characterizes and controls the chemical nature of each lipid. Additionally, we allow coupling with collective variables and show that it can be used to dynamically create asymmetric membranes. This algorithm is openly available as a plugin for the HOOMD-Blue molecular dynamics engine.

Significance

We present a software solution for molecular dynamics simulations in which the populations of molecular species are controlled by chemical potentials. An efficient implementation in the semigrand canonical ensemble is presented, with applications to phospholipid regulation in membranes. Coupling the chemical-potential values to collective variables facilitates the simulation of asymmetric membranes.

Introduction

Membranes in eukaryotic cells are mostly composed of lipids with particularly complex chemistry and organization. A typical mammalian cell contains hundreds of different types of lipids within any one of its membranes, distributed asymmetrically between both leaflets (1,2). The chemical nature of most lipids—headgroup composition, acyl tail length, degree of saturation—is maintained by the Lands’ cycle in the endoplasmic reticulum. The asymmetric distribution is maintained by membrane-embedded type IV P-type ATPase (P4-ATPAse) proteins, also known as flippases, which consume ATP to move lipids from one leaflet to the other. Why cells require such a complex distribution of lipid chemistries along with a tight regulation remains an open question. Computer simulations have proven valuable to garner insight into the behavior of simple model membranes and are moving toward realistic biological chemistry (3). For instance, the MARTINI model (4) has been used to model realistic simulations of plasma membranes (5). However, understanding the fundamental reasons for membrane composition and asymmetry requires systematic variations of the myriad of potential compositions. Moreover, simulations involving asymmetric compositions must be executed carefully so as to properly account for possible differential stress (6).

How do membranes regulate their composition? Giant plasma membrane vesicles, extracted from plasma membranes that retain composition, are necessarily void of any biochemical regulation process and thus retain a fixed composition. They are known to possess a miscibility transition temperature just under cell growth temperature (7) and display apparent critical behavior (8). These results clearly show that lipid composition responds to environmental changes. Computer simulations remain unable to correctly model regulation, which involves chemical reactions and intermembrane lipid diffusion. Similarly, regulation is difficult to characterize experimentally: the precise control of many types of lipids requires a sophisticated sensing mechanism that remains largely elusive. We recently hypothesized that phospholipids may follow a loose regulation mechanism, thermodynamic sensing via the chemical potential, whereas other components such as cholesterol are tightly regulated (9). These regulated ensembles correspond to mixtures of canonical and semigrand canonical (SGC) ensembles, that is, the chemical-potential difference between molecules within a set is fixed, and the number of molecules in each set is also fixed. Practically, some components may change their chemical nature over time, whereas their overall number is constrained. This scenario naturally and robustly self-regulates toward critical points (10).

Here, we present the software we previously employed to simulate regulated lipids in SGC ensembles (9). Highly parallel algorithms for SGC already exist (11), available in multiple molecular dynamics packages, e.g., LAMMPS and openMM (12). Our contribution addresses two previously missing features specific to membrane simulations. First, the enforcement of chemical potentials must be made at the level of entire molecules. Second, modeling heterogeneous scenarios requires the use of collective variables, for instance, to assign different chemical potentials to each leaflet of the membrane. To this end, we associate a chemical potential to any arbitrary combination of chemistry, charge state, and collective variable. Our implementation, which runs on graphical processing units, is available as a plugin for the HOOMD-Blue molecular dynamics engine (13,14). We demonstrate this implementation on a lipid bilayer with an asymmetric composition. We postulate that P4-ATPAse induces a chemical-potential difference that only depends on headgroup nature (phosphatidylcholine (PC) versus phosphatidylethanolamine (PE)). This allows us to relate the work done by P4-ATPAse to dynamically create and stabilize an asymmetric membrane profile.

Methods

Following Sadigh et al., the algorithm employed here makes use of a simulation-domain checkerboard decomposition (see Fig. 1 A) (11). The simulation box is decomposed into cells of minimal thickness σ, where σ is the largest interaction range in the system, e.g., σ ≈ 1.2 nm for the MARTINI force field. Particles located at least two cells away from each other are therefore noninteracting. Every update step, the algorithm selects a set of noninteracting cells (depicted in blue in Fig. 1 A). In each active cell, one particle is randomly selected and a swap is attempted, with acceptance determined by the usual Metropolis criterion exp(−β(ΔU − Δμ)), where β = (k_BT)⁻¹ is the inverse temperature, ΔU is the internal energy change, and Δμ is the chemical-potential difference between the two species.

Figure 1.

Schematic of the SGC algorithm employed. (A) 2D representation of the checkerboard decomposition for a lipid membrane; lipids are drawn and colored in MARTINI representation, and the checkerboard and active cells are shown in green and blue, respectively. A particle is chosen randomly within each active cell for an alchemical transformation. The pathological molecule is shown in red stretches across multiple active cells, which can lead to data races. (B) Calculation of the molecule hash for a typical molecule in a bilayer, with color indicative of SGC representation. Each bead in the molecule corresponds to an offset in the hash, facilitating local bit changes. For instance, an alchemical transformation of bead labeled 5 from green to gold (state 0–1, respectively) results in a change of the hash on bit 5 only. The last, leftmost bit controls a discrete, finite-valued collective variable, for instance, leaflet side. To see this figure in color, go online.

To associate a chemical potential to a given molecule, we assign a unique number, i.e., a hash, to each chemical structure. Additionally, we can include information about the environment, e.g., the leaflet of the membrane, by means of discrete-valued collective variables. The hash simply aggregates all possible chemical states that a molecule may form. As an example, let us consider the coarse-grained lipid depicted in Fig. 1 B, which relies on the MARTINI force field (4). We allow seven beads to change their chemical type (shown in color and numbered on Fig. 1 B). First, the headgroup (red) can change between PC and PE, represented by MARTINI bead types Q₀ and Q_d, respectively. The beads shown in green can be either saturated or unsaturated, with bead types C₁ and C₃, respectively. The last bead, in blue, not only controls (un)saturation but also the length of the acyl tail via a “ghost” (i.e., empty) particle. These particles interact through a normal bonded force field and therefore have well-defined mass and momenta. However, they have zero nonbonded interactions, which effectively makes them invisible. This is chosen to avoid “true” particle insertion, which tends to be numerically inefficient. The hash is constructed from a minimal binary representation: a two-state or a three-state bead occupies one or two bits, respectively. This leaves some hash values to point to unphysical states, e.g., there are more than three states in two bits. Additionally, some states may be chemically unavailable, e.g., noncontiguous unsaturations in biologically relevant lipids. Both unphysical and chemically unavailable states are assigned a chemical potential μ = −∞ to forbid any alchemical transformation. Chemical potentials are user assigned for every hash.

The Monte Carlo procedure described in the Appendix, Algorithm 1 extends Sadigh et al. (11) but requires a few more memory transactions. After choosing a random particle, the algorithm must resolve the molecule it belongs to, as well as its hash. A new state and corresponding hash are then randomly generated. This procedure requires resolving the hash offset of the current bead. Because of the parallel nature of the transformations, large molecules spanning multiple active cells can result in hash data races (see red molecule in Fig. 1)—concurrent read or write commands from multiple threads, resulting in corrupt data states. To remedy this situation, a molecular lock prevents multiple changes on the same molecule within a single Monte Carlo step.

The base-2 representation of the chemical states allows for bitwise operations, ensuring high numerical performance. This comes at the cost of the extent of chemical space that can be described by a single hash; for instance, in Fig. 1 B, the terminal blue bead uses two bits to represent three chemical states (coupling and unsaturation). This leaves one of the four two-bit states left unassigned. Because we use 32-bit integers for hashes, the worst-case scenarios involve either losing a chemical state every two bits (e.g., a molecule composed of only blue bits in Fig. 1), leading to 3¹⁶ = 4.3 × 10⁶ chemical species or using beads with more than 2¹⁶ + 1 = 65,537 chemical states, in which case there can be only a single such bead per molecule. Although this may appear as limited compared to the maximal number of states described by 32 bits (2³² = 4.3 × 10⁹ states), this is largely sufficient for all practical purposes.

The incorporation of collective variables allows us to control larger-scale behavior, for instance, compositional asymmetry. Chemical potentials that are different on the two leaflets of a membrane will generally lead to an out-of-equilibrium situation. These systems can exhibit peculiar properties, such as net flows, that tend to depend on the system’s kinetic rates. To alleviate artifacts, the natural relaxation timescale of the system (e.g., flip-flop for asymmetric phospholipids bilayers) must be much longer than the total simulation time. Otherwise, the system will depend on simulation conditions and, in particular, on the SGC relaxation timescale.

Results

To demonstrate the value of our method, we examine a biologically relevant system: a membrane with an asymmetric lipid composition. We simulate a membrane comprised of PC and PE lipids. On the upper leaflet, we impose a difference between PC and PE molecules of Δμ (which we will vary) as a proxy for the effects of P4-ATPAse proteins. As outlined in the Introduction, this assumes that P4-ATPAse binds all PC molecules equally, independent of acyl tail nature. On the lower leaflet, we fix Δμ = 0 between the two chemical species.

We measure asymmetry via the parameter δ^± = $\left(N_{\text{PC}}^{\pm }-N_{\text{PE}}^{\pm }\right)$/N, which probes differences in headgroup population on each leaflet (see Fig. 2). As expected, δ⁺ shows a sigmoid-like behavior, in which the free energy is largely dominated by the mixing entropy at large values of Δμ. The symmetric configuration δ⁺ = 0 occurs away from Δμ = 0 because of hydrogen bonding between PE heads. The composition of the lower leaflet barely changes, i.e., δ⁻ is flat, and corresponds to ∼88% of PE lipids. The stark difference between the two curves highlights the absence of coupling between headgroup composition in the two leaflets. The two curves intersect at Δμ = 0, as expected.

Figure 2.

Asymmetric membrane properties. (A) Snapshot of a typical configuration at Δμ = 10 kJ/mol. Cholesterol is colored in white, and PC and PE are colored according to their unsaturation level on different color scales to differentiate them. (B) Resulting headgroup asymmetry δ^±. At Δμ = 0, PE molecules dominate both layers, with ∼88% of molecules being PE. The composition of the lower leaflet is nearly unaffected by the changes of the upper leaflet. To see this figure in color, go online.

Beyond the stabilization of asymmetric membranes, this simulation also yields an important result: P4-ATPAse proteins need to exert $\underset{˜}{\gtrsim }$20 kJ/mol of work on lipids to create a strongly PC-dominated upper leaflet. This value is compatible with the free energy made available during ATP hydrolysis (∼30 kJ/mol).

The number of Monte Carlo attempts per step grows linearly with system size, leading to an overall time complexity $\mathcal{O}$(NlogN). This favorable scaling makes the algorithm amenable to study large-scale systems, for instance, for finite-size scaling of critical membranes. Additionally, if the membrane has only a single SGC ensemble with all components being regulated—unlike, say, cholesterol—equilibration becomes independent of long-range diffusion. The situation is akin to molecular dynamics coupled with alchemical steps, in which two lipid chemical states are swapped in composition-conserving nonequilibrium transformations (15,16). However, such a scheme can only attempt a single nonequilibrium move per update, which in turn implies a much less favorable $\mathcal{O}$(N²log(N)) time complexity.

In principle, the algorithm is amenable to all chemistries. However, it suffers from the traditional Monte Carlo drawbacks; particle insertion required for lipid elongation is typically problematic. At a rate of one sweep every 200 fs, this results in equilibration times for lipid length on the order of ∼1 μs, whereas saturation has equilibration times on the order of 10 ns. For simulations incorporating nonregulated components, e.g., cholesterol in membranes, one has to factor in the relaxation of nonregulated components (∼1 μs for MARTINI cholesterol). For this application, using the MARTINI force field, this is therefore not a problem. However, for atomistic simulations, this will likely be problematic. Additionally, our implementation only performs single changes per bead per step and precomputes all potential alchemical transformations. Accordingly, the computational cost of a time step in which a Monte Carlo update is performed is increased by ∼60%. However, because updates are seldom performed, e.g., 3 out of 20 timesteps for the membrane shown in Fig. 2, the performance penalty is generally on the order of ∼10% when measured by HOOMD benchmark utilities.

Conclusions

We present an implementation for molecular SGC, facilitating membrane simulations with lipids regulated by chemical potentials. The inclusion of collective variables in the hash representation facilitates the stabilization of different (asymmetric) leaflet compositions. We hope that the tools deployed here will assist simulation research on regulated ensembles and asymmetric membranes.

Data and code availability

Molecular dynamics simulations make use of the HOOMD-Blue engine (13,14,17), a DPD thermostat (18), and the MARTINI force field (4). Initial topologies are built using the hoobas molecular builder (19). The SGC HOOMD-Blue plugin for HOOMD-version 2.9.3 is available in the Supporting materials and methods, as well as in https://gitlab.mpcdf.mpg.de/mgirard/SGC-molecules. The repository also contains the input file to run the example described in this work. Documentation is available at https://sgc-molecules.readthedocs.io/en/latest/.

Author contributions

M.G. and T.B. designed the research. M.G. wrote the software, carried out simulations, and analyzed the data. M.G. and T.B. wrote the article.

Editor: Markus Deserno.

APPENDIX: ALGORITHM 1 MONTE CARLO PROCEDURE

for c in active cells do

• $p\leftarrow \text{random particle}\in c$

• $\text{mol}\leftarrow \text{MoleculeIndex}\left[p\right]$

• $\text{molHash}\leftarrow \text{Hash}\left[\text{mol}\right]$

• $o\leftarrow \text{Offset}\left[p\right]$

• $s\leftarrow \text{States}\left[p\right]$

• $s^{\prime }\leftarrow \text{Random state}\ne s$

• $\text{molHas}{\text{h}}^{\prime }\leftarrow (\text{molHash}\&\sim (\text{mask}\ll o))|(s^{\prime }\ll o)$

• $\text{Δ}U\leftarrow U\left[s^{\prime }\right]-U\left[s\right]$

• $\text{Δ}\mu \leftarrow \mu \left[\text{molHas}{\text{h}}^{\prime }\right]-\mu \left[\text{molHash}\right]$

• $\mathbf{if}R\left(\mathrm{0,1}\right)<\text{exp}\left(-\beta \left(\text{Δ}U-\text{Δ}\mu \right)\right)\mathbf{then}$

  • $\text{lock}\leftarrow \text{atomicCAS}\left(\&\text{Lock}\left[\text{mol}\right],\mathrm{0,1}\right)$
  • if lock then return
  • $\text{Hash}\left[\text{mol}\right]\leftarrow \text{molHas}{\text{h}}^{\prime }$
  • $\text{States}\left[p\right]\leftarrow s^{\prime }$