Bridging the gap between chemical processes and the mesoscopic dynamics of artificial cells requires a simulation methodology able of interpolating between MD and length scales on which hydrodynamics plays a role. Dissipative particle dynamics (DPD) is a now well-established particle-based approach in the field of soft-matter sciences, and we derived extensions meeting the specific needs of simulations reflecting the multi-scale chemical and morphological dynamics of artificial cells.
This approach has born fruit in the simulation of the entire life-cycle of two types (micellar and vesicular) of artificial cell.
There was, however, a fundamental difficulty to overcome: It was originally stated that the method is scale-free. This means that the parameters used in the simulation do not depend on the level of coarse-graining [1]. In a later publication [2], this earlier finding was declared erroneous. Based on the scaling relation by these authors, the performance of DPD was analyzed for various coarse-graining levels [3],[4]. It was found that there exists an upper coarse-graining level above which the simulated fluid freezes. Trovimof reported that this coarse-graining limit is disappointingly low [4], and only allows up to about 10 water molecules to be grouped together into one DPD particle. This limit would prevent DPD from covering the whole mesoscopic range and confines its applicability essentially to the order of magnitude of MD simulations. In [5], it was argued that the original statement that DPD is scale free can be perpetuated, provided one uses an appropriate scaling scheme differing from that described by Groot and Rabone. The scaling we employ preserves the compressibility of the system under coarse graining, takes into account the decrease of the number of degrees of freedom in a way that directly preserves the temperature. Additionally, by exploiting the inherent gauge freedoms of the DPD-algorithm, it was shown that a given DPD-simulation represents the dynamics of a whole family of physical systems parameterized by length- and time scales and thereby DPD was re-established as an inherently scale-free method, also suited for describing phenomena on scales much above the one of individual molecules.
References
1. R. D. Groot and P. B.Warren, J. Chem. Phys. 107 (11), 4423 (1997).
2. R. D. Groot and K. L. Rabone, Biophysical Journal 81 (2001), 725.
3. W. Dzwinel and D. A. Yuen, J. Mod. Phys. C 11 (2000)(1), 1.
4. S. Trofimov, Thermodynamic consistency in dissipative particle dynamics, Ph.D. thesis, Technische Universiteit Eindhoven (2003).
5. R. M. Füchslin, H. Fellermann, A. Eriksson and H.J. Ziock, “Coarse-Graining and Scaling in Dissipative Particle Dynamics”, http://arxiv.org/abs/cond-mat/0703682
Recent development in nano- and bio-technology has brought increased attention to the need for simulations that can accurately describe mesoscale systems. Various methods have been suggested in the literature, but it is clear that simulations with similar accuracy as we have on smaller scales (e.g. molecular dynamics, MD) and larger scales (e.g. Navier-Stokes equations), are still lacking for mesoscopic systems. The problem is not only a matter of length and time scales. It is often easier to understand the systems behavior in the low or high temperature limits. The systems we are interested in, however, consist of a large (but not huge) numbers of heterogeneous molecules, interacting at a temperature where significant contribution to the free energy comes both from the potential energy and the entropy (this is sometimes referred to as kBT-physics). A major complication is that the dynamics is sensitive to the structural details of the molecules involved, but at the same time it is too computationally expensive to run a simulation with atomic-scale resolution. We must search for a coarse-graining scheme that can combine pertinent atomic-scale information, and at the same time be computationally efficient enough to reach the length and time scales on which molecular self-assembly occurs. Ideally the simulation should also be able to describe systems out of equilibrium.
Coarse-grained Molecular Dynamics (MD) simulation methods could, if further developed, provide simulation tools that can address phenomena occurring on time and length scales that are not reachable directly using standard MD. At the same time they can capture details of the molecular structure that continuum models cannot. United atoms (UA) [1] and dissipative particle dynamics (DPD) [2] are currently among the leading candidates for such methods. Both have been used to provide insight into a variety of phenomena that occurs at the mesoscale, for example complex fluids [3], biomembranes [4], and lipid bilayers [5]. In the near future it is expected that UA and DPD will play an increasing role as multi-scale modeling approaches to bridging atomistic and continuum descriptions.
The DPD method finds its formal motivation in the Mori-Zwanzig theory for projection operators [5,6], and in that context it has received considerable theoretical attention, e.g. [7,8,9]. However, a full understanding of the relation between the macroscopic dynamics and the mesoscale representation is still lacking. For example, the functional form of the stochastic interactions is typically chosen ad hoc, after which the overall interaction strength is tuned to match global thermodynamic observables. It is well known that the stochastic interactions strongly affect the transport properties in bulk fluids [12,13]. Correct representation of the stochastic interactions in the water surrounding the macromolecules is a necessity for realistic modeling where the time-scales of the transport processes are important, e.g. modeling the dynamic states of biomembranes, and transport in and across membranes. Furthermore, establishing systematic methods, backed by established theory, to derive mesoscopic simulation methods is necessary for bridging the gap between MD and continuum hydrodynamics.
The larger picture: the assembly of a proto-cell
Generally, our motivation for developing accurate and efficient simulations methods is to understand the dynamics of self-assembling bio-chemical systems. These systems play a central role in living organisms, especially in connection to bio-membrane dynamics. Understanding the detailed dynamics in such systems would constitute a cornerstone in our knowledge of e.g. inter- and intra-cellular transport, morphogenisis, and membrane function. On the engineering side, an important lesson can be learned from fluid mechanics: accurate simulation tools are primary drivers for efficient technology development. There are reasons to believe that the same holds true in nano-technology: as nano-fabrication continues to migrate from the basic science sphere into the field of engineering, we expect an increased demand for reliable computational tools for nano- and meso-scale systems. The Programmable Arificial Cell Evolution (PACE) project defines a new research program residing at the boarder between nano- and bio-science: chemical implementation of artificial proto-cells. The project relies to a large extent on “exploratory chemistry†and it is primarily in this context that the novel simulation methods we develop are essential.
Simulation of self-assembly processes
Traditionally, theoretical and computational studies of self-assembling systems have mainly been performed by microscopic lattice models, Ginzburg-Landau theories, membrane theories (thin shells), simple lattice gas models, and lately also by molecular dynamics (MD) simulations. Many of these models have been successful in explaining qualitative experimental results (mainly producing topologically correct phase diagrams). One of the most successful techniques has been thin shell models where the shape transformations of the self-assembled structure can be modeled using Hamiltonian dynamics where bending energy and surface tension are captured as effective potential energies. The Hamiltonian gives an effective representation of the conservative and entropic forces in the system, i.e. the free energy. However, all these models, except molecular dynamics, are intrinsically unable to faithfully represent the structural details of the interacting objects; water, lipids and amphiphilic molecules. Many aspects of both the dynamics and the equilibrium depend strongly on the detailed structure of the molecules and their interactions: water molecules form hydrogen networks, the length of the lipid polymer determines the stability of assembled structures, the strength and geometric properties of the interaction between amphiphilic molecules and water can directly affect the size distribution of micelles, stability of membranes, etc. Molecular dynamics (MD) provides a powerful tool for studying bilayers and vesicles. However, computational constraints limit the applicability of MD to relatively small systems and/or short time scales. The spontaneous formation of vesicles in aqueous environments can for example not be simulated with MD.
Particle based mesoscale simulation techniques
There is a hope that the gap will be closed by particle based mesoscale simulation methods, such as dissipative particle dynamics (DPD). DPD has been used to simulate spontaneous micelle and vesicle formation of amphiphilic molecules in aqueous environments, e.g.[14,15]. The water surrounding the self-assembling amphiphiles is coarse grained so that collections of water molecules are represented by coarse grained mesoscale particles, i.e. a DPD representation. The amphiphilic molecules are coarse grained by splitting the polymer into a hydrophobic tail groups and hydrophilic head groups, i.e. a UA representation. The different parts of the amphiphilic molecules are represented by UA particles with different effective interactions depending on their hydrophobicity. The internal covalent bonds are modeled using spring like interaction potentials. Overall the coarse graining is performed so that all the UA and DPD particles, i.e. water and parts of the amphiphilic molecule, have approximately the size of a CH2-group. The interactions between the coarse grained particles are modeled by a soft-core repulsive potential. The functional form of the interactions are chosen heuristically and calibrated using macroscopic obsevables. This method is unsatisfactory and we will discuss improvements below.
In a model presented in [14,15] and in Fig. 1, the spontaneous formation of vesicles from an initial state of randomly distributed amphiphilic molecules is studied. The self-assembly process undergo a number of distinct stages. Initially the amphiphilc molecules cluster together into micelle structures (see Fig. 1); the micelles then grow and reforms into oblate geometries or a bilayer membrane, depending on the concentration of amphiphilc molecules; the flat structures, membranes or oblate micelles, have large longetudal fluctuations and after some time they close and form vesicles surrounding a core of water. At high density the formation of rod-like structures is also observed. While the results presented in Fig. 1 are qualitatively correct, there are several serious shortcomings in the model:
- The effective interactions are decided heuristically and there is no general method for adopting the model to represent other types of amphiphiles etc.
- The DPD particles representing water cannot diffuse through the membrane. This is especially problematic in the vesicle phase since the pressure gradient through the membrane cannot be equilibrated - clearly an unrealistic feature of the model. This can also be seen in the last picture in Fig. 1 where the vesicle never reaches a spherical shape.
where Fij is the force between particle i and particle j, and Fcij is the conservative part of the force. Note that the average is conditioned on the distance between the particles rij, whereby we get a radially dependent stochastic force interaction. The consistency of the method depends on the existence of a region in time, the time-scale of integration of the slow dynamics. Using data from a detailed MD simulation, and using the equation above, the strength of the central stochastic interaction between particles at distance r can be determined. The dissipative interaction is then also established through the fluctuation-dissipation theorem.
Summary
We have developed a method that can be used to estimate the effective pairwise Langevin dynamics for mesoscopic simulation models. It is a bottom up method based directly on projection operator methods in non-equilibrium statistical mechanics. The main point of using correct stochastic interaction in mesoscale simulations is to accurately capture fluctuations from equilibrium and transport properties in a coarse grained representation. Our results demonstrate that it is possible to have relatively good agreement between transport properties in a detailed molecular dynamics simulation and a coarse grained model based on a DPD thermostat in two different systems: a Lennard-Jones fluid [18] and a Single Point Charge water model [19].
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