Particle-based membrane model for mesoscopic simulation of cellular dynamics
Summary (4 min read)
- The authors expect their model to be of high practical usability for ultra coarse-grained molecular dynamics or particle-based reaction-diffusion simulations of biological systems.
- Bilayer membranes have been the subject of computer simulations for more than three decades.33–35.
- Their approach consists of calibrating the interaction potentials to result in desired macroscopic mechanics and structural properties.
- The proposed model is essentially an elastic membrane model, comparable to triangulated models, with the difference that the desired elastic properties are reproduced through simple bonded interactions in contrast to complicated orientation- or curvature-dependent potentials.
II. THE MODEL
- As shown in Fig. 1(a), two close-packed lattices of particles correspondingly represent the two leaflets of the membrane in this model.
- The elastic energy density contributed to the membrane is usually expressed in terms of the local curvature of the mid-surface of the bilayer.
- The authors aim to avoid computing complex potential functions based on numerically obtained local curvature values.
- Thus, only bond-stretching and anglebending interactions amongst nearest neighbor particles are considered.
- Considering an arbitrarily curved membrane, and based on its local surface geometry, relative configuration of particles and the resulting bond lengths and angles are obtained.
B. Differential geometry of the particle-based membrane model
- In which the membrane is effectively composed of “particle dimers,” i.e., pairs of particles belonging to the top and bottom leaflets, a hypothetical mid-surface is assumed to lie halfway between the particle dimers.
- Inspired by classical continuum shell theories, the authors assume that bending of the double layer deforms it such that a normal vector originating from a point p on the mid-surface, pointing to a particle P on the upper or lower layer, remains perpendicular to the mid-surface, independent of the state of deformation [see Fig. 1(b)].
- Without loss of generality, the authors focus on particles positioned on the top leaflet for the following derivations.
- For the purpose of calculating partial derivatives of the normal vector, Weingarten’s formula, n,µ = −bνµeν = −bµνgνσeσ has been used.
- Up to this point, the derived equations hold in all local coordinate systems at point p. A smart choice of the coordinate system can simplify the equations considerably.
C. Parameter-space optimization of interaction potentials
- Now that the authors have obtained equations describing the relative configuration of model particles in an arbitrarily curved membrane [Eqs. (4) and (5)], they can select interaction potentials which are functions of |rPQ | and θpPQ and calculate effective energy densities corresponding to arbitrary curvature states.
- With such a definition for area elements, half of each lateral bond emanating from a particle P, plus all the out-of-plane angles having it as the vertex, is included in one area element around particle P.
- But without performing the simulation, the authors do not have a priori knowledge of the in-plane angle, ψ, that this starshaped construct around each particle makes with the principal directions of the curvature of the mid-surface.
- Thus, in general, the calculated effective energy density depends on this in-plane angle.
- This way, the effective potential energy density is defined as feff = 〈 1 2 ∑ Ustretch + ∑ Ubending ∆A 〉 ψ , (10) where the summations are carried out for all interactions corresponding to one particle and ∆A denotes the area element.
D. Additional interactions
- The bond-stretching and angle-bending interactions described in Sec. II C only serve to reproduce the desired curvature elasticity of the membrane.
- In addition to these interactions, other potentials can be added to the model for different geometrical or mechanical considerations, as long as they do not perturb the effective energy density described by Eq. (10).
- Most importantly, a harmonic potential is added between particles in each dimer.
- Addition of this potential is necessary to hold the two leaflets together.
- Also, with the set of interactions described so far, volume exclusion is only present between neighboring membrane particles, and in principle, non-neighboring particles can interpenetrate.
E. Bond-flipping moves
- In contrast, lipid bilayer membranes are two-dimensional fluids in which lipid molecules can freely diffuse laterally, and this fluidity is essential for membrane remodeling.
- 84 Following a scheme commonly used in triangulated membrane models,70,71 the in-plane fluidity is introduced to the model via bond-flipping Monte Carlo moves.
- This proposed move is accepted with the Metropolis-Hastings probability of exp [−β (Enew − Eold)] where Eold and Enew are the corresponding potential energies of the system in the old and new topologies, and β = 1/kT with k being the Boltzmann constant and T the temperature.
- In a simulation in the canonical ensemble, this lost energy will be compensated by the thermostat, which is the same as extracting work and adding equal amount of heat to the system.
- The frequency, φ, with which the bond-flipping moves are proposed, acts as a control parameter for the model.
B. Time integration
- In order to simulate tensionless membranes in thermal equilibrium, an extended system dynamics approach is used to derive equations of motion and devise the proper numerical integration scheme.
- 92–94 Thermostatting is achieved through NoséHoover chains, and isotropic cell fluctuations are used for barostatting to achieve zero in-plane tension.
- On the other hand, in the absence of any solvent effects, and with the deterministic MTK integrator used here, the out-of-plane dynamics is solely determined by the particle masses and the stiffness of the forcefield developed based on the scheme introduced in Sec. II C.
- As the forcefield is the outcome of the parameter-space optimization aiming to reproduce the desired membrane elasticity, the only remaining parameter is the mass of model particles.
- Achieving physically relevant out-of-plane dynamics pertaining to membrane patches suspended in a solvent is only possible through either implementing a suitable stochastic integrator or including solvent effects explicitly or implicitly.
C. Simulation code and visualization
- Mainly due to the fact that the implementation of bondflipping Monte Carlo moves in available molecular dynamics software packages proved impractical, an in-house C++ code has been developed to handle the simulations.
- Visualization is done via the Visual Molecular Dynamics (VMD) software package.95.
A. Thermal undulations
- A lipid bilayer patch in thermal equilibrium undergoes significant out-of-plane thermal undulations.
- These undulations can be studied from a statistical mechanics point of view to obtain energy distribution among different vibration modes.
- To ensure that the membrane patches have indeed been equilibrated, an estimate of the relaxation time of the system is required.
- The parameters for these fits are given in Table II.
- It is observed that increasing the lattice parameter in general has little effect on the ability of the model to reproduce continuum behavior.
B. Area compressibility
- As an example of additional physical properties of the model that can be taken into account when choosing potential parameters, area compressibility of the membrane is calculated for a model for which potential parameters are chosen from different families.
- The results are ordered as functions of the stiffness of the angle-bending potential, Kb, and are depicted in Fig.
- The fluidity of the 2D liquid is described in terms of the surface viscosity, which arises from the assumed linear relation between the in-plane shear stress and the corresponding velocity gradient.
- Thus, the general procedure described in this section has to be repeated if another integrator is used.
D. Nanoparticle wrapping
- As a final test of the usefulness of their membrane model to handle substantial deformations and model biologically relevant membrane remodeling processes, the authors simulate the interaction of spherical nanoparticles with the membrane, as a well-known benchmark system.
- It is a useful test for the membrane model to show that (a) the model offers enough flexibility to simulate the budding behavior of bilayer membranes, and (b) if it correctly reproduces the interplay between bending and adhesion energies.
- For this type of nanoparticle-membrane interaction, and with a continuum membrane model, semianalytical studies105 have been carried out on the degree to which the surface of the nanoparticle is covered by the membrane, as a function of the dimensionless adhesion energy u = UpR2/κ as well as the potential range, ρ.
- It is observed that the model follows this prediction with very good accuracy.
- The good fit to the catenary curve is an indication that the particle-based model very well captures the zeroenergy regions and assumes corresponding minimal surface geometries.
- The authors have described a strongly coarse-grained model for simulating lipid bilayer membranes that is similar in nature with triangulated surface models, but is purely particle based, and as such is suitable for seamless integration into interactingparticle reaction-diffusion simulations.
- The model relies on bond-stretching and anglebending interactions among nearest-neighbor particles with parameters optimized to reproduce prescribed macroscopic curvature elasticity.
- These computer experiments have proven the model to be reliable in different equilibrium and non-equilibrium simulations and correctly predict the expected behavior of lipid bilayer membranes as two-dimensional fluids obeying curvature elasticity.
- Armed with these capabilities, the authors ultimately aim to use this coarse-grained model in the context of iPRD simulations to study cellular signal transduction at large spatiotemporal scales.
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Frequently Asked Questions (2)
Q1. What future works have the authors mentioned in the paper "Particle-based membrane model for mesoscopic simulation of cellular dynamics" ?
Armed with these capabilities, the authors ultimately aim to use this coarse-grained model in the context of iPRD simulations to study cellular signal transduction at large spatiotemporal scales.
Q2. What are the contributions mentioned in the paper "Particle-based membrane model for mesoscopic simulation of cellular dynamics" ?
In this paper, a particle-based model for simulating lipid bilayer membranes is presented, which is similar in nature with triangulated surface models, but is purely particle based, and as such is suitable for seamless integration into interactingparticle reactiondiffusion simulations.