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Self-Assembly of Nanocomponents into
Composite Structures: Derivation and Simulation
of Langevin Equations
Article in The Journal of Chemical Physics · June 2009
DOI: 10.1063/1.3134683 · Source: PubMed
4 authors, including:
Stephen Pankavich Zeina Shreif Colorado School of Mines National Institutes of Health 39 PUBLICATIONS 191 CITATIONS 22 PUBLICATIONS 220 CITATIONS SEE PROFILE SEE PROFILE Yinglong Miao Indiana University Bloomington 45 PUBLICATIONS 443 CITATIONS SEE PROFILE Available from: Zeina Shreif Retrieved on: 17 October 2016 Self-Assembly of Nanocomponents into Composite Structures: Derivation and Simulation of Langevin Equations S. Pankavicha,b,*, Z. Shreifb, Y. Miaob, P. Ortolevab Department of Mathematicsa University of Texas at Arlington Arlington, TX 76019 Department of Chemistryb Center for Cell and Virus Theory Indiana University Bloomington, IN 47405 * Corresponding Author – S. Pankavich email@example.com Abstract The kinetics of the self-assembly of nanocomponents into a virus, nanocapsule, or other composite structure is analyzed via a multiscale approach. The objective is to achieve predictability and to preserve key atomic-scale features that underlie the formation and stability of the composite structures. We start with an all-atom description, the Liouville equation, and the order parameters characterizing nanoscale features of the system. An equation of Smoluchowski type for the stochastic dynamics of the order parameters is derived from the Liouville equation via a multiscale perturbation technique. The self-assembly of composite structures from nanocomponents with internal atomic structure is analyzed and growth rates are derived.
Applications include the assembly of a viral capsid from capsomers, a ribosome from its major subunits, and composite materials from fibers and nanoparticles. Our approach overcomes errors in other coarse-graining methods which neglect the influence of the nanoscale configuration on the atomistic fluctuations. We account for the effect of order parameters on the statistics of the atomistic fluctuations which contribute to the entropic and average forces driving order parameter evolution. This approach enables an efficient algorithm for computer simulation of self-assembly, whereas other methods severely limit the timestep due to the separation of diffusional and complexing characteristic times. Given that our approach does not require recalibration with each new application, it provides a way to estimate assembly rates and thereby facilitate the discovery of self-assembly pathways and kinetic dead-end structures.
Keywords: self-assembly, coarse-graining, multiscale analysis, nanoscience, Liouville equation, bionanostructures, viruses, ribosomes
Self-assembly is the natural and spontaneous organization of simple components into larger patterns or structures without human intervention. This phenomenon occurs frequently within nature and technology and can involve components from a variety of scales, from the molecular to the macroscopic1,2. In this article, the self-assembly of a composite structure from nanoscale components is analyzed using a multiscale approach. Biological systems that self-assemble for which the present approach is designed include the viral capsid, ribosome, and cytoskeleton.
Opal is a geological composite, and engineered composite materials have great promise as short materials. The self-assembly of these systems typically takes place on millisecond or longer timescales. As atomic collisions and vibrations occur on the 10-14 second scale, their collective influence drives self-assembly while their dynamics are simultaneously affected by the slower processes. The fast processes act at the atomic scale, whereas those at the nanoscale involve the coherent motion of thousands or more atoms simultaneously. Thus, from both the temporal and spatial perspectives, self-assembly has multiscale character. The objective of this study is to show how laws of self-assembly can be derived via a multiscale analysis of the basic laws of molecular physics (notably the Liouville equation) for systems describable as a set of classical atoms evolving under the influence of an interatomic force field. More specifically, we rigorously derive an equation for the stochastic dynamics of variables describing the selfassembling nanocomponents and show how this development leads to a theory free from recalibration with each new application.
It is envisioned that the theory developed here will provide a framework for analyzing a
variety of self-assembly phenomena, including:
• dimerization and the formation of other protein complexes • formation of viruses, ribosomes, and other bionanostructures • construction and loading of nanocapsules for drug, gene, or siRNA delivery • creation of macromolecular circuits or cytoskeletal structures • formation of engineered composite materials • creation of geological composites, such as opal.
Viral capsid self-assembly is of particular interest to the medical and engineering industries.
Antiviral strategies have been proposed with the aim of interfering with the growth of viral infection by targeting the assembly of viruses using antiviral therapeutics3. The self-assembly mechanism of viral capsids has also been applied to synthesize functionalized supramolecules4 which can then be utilized as molecular containers for engineered nanomaterial synthesis5-9.
Such self-assembling systems are key aspects of great scientific and technical interest, so that a conceptual and computational advance in self-assembly theory could have a broad, practical impact.
In this study, we focus on self-assembly of objects from nanocomponents, such as viral capsids from capsomers or opal from silica spheres (although its formation is not self-limiting).
In these cases, the assembling nanocomponents each consist of many atoms so that the behavior of individual components has mixed atomic-chaotic and coherent character. Such mixed behavior systems have the character of Brownian motion. Therefore, the evolution of such a self-assembling system can be described as the result of interscale cross-talk. Atomic fluctuation provides the entropies for free energy driving forces, as well as stochastic forces to overcome energy barriers and create Brownian motion. Conversely, the coherent aspects (order parameters) of these systems, notably their nanoscale architecture, modify the statistical proportions of the atomistic fluctuations. This interscale cross-talk creates the feedback loop
Three aspects of self-assembling systems of specific interest are the following:
(1) The assembly self-limits its size, in contrast to precipitation wherein growth of a solid is only limited by the number of available components.
(2) The structures are hierarchical both in their architecture (atoms make proteins, proteins make capsomers, and capsomers make viral capsids) and in their growth kinetics (small subunits form substructures which then assemble into more extensive ones).
(3) These systems can best be understood via an analysis that integrates processes communicating across multiple scales in both space and time.
Multiscale analysis is a way to study systems that simultaneously involve processes on widely separated time and length scales. It has been of interest since the work on Brownian motion by Einstein10-21. In these studies, Fokker-Plank (FP) and Smoluchowski equations are derived either from the Liouville equation or via phenomenological arguments for nanoparticles without internal atomic-scale structure. Recently, we improved this work by accounting for atomic-scale internal structure, introducing general sets of structural order parameters characterizing nanoscale features of the system, establishing a way to include these variables in the analysis without the need to track the number of degrees of freedom, and incorporating specialized ensembles constrained to fixed values of the order parameters to construct the average forces and friction coefficients in the equations of stochastic order parameter dynamics22-28.
Advances in the theory of chemical kinetics are relevant to self-assembly. Multiscale analysis of the Liouville equation for reacting hard spheres11,12,14 shows that when the probability for reactive collision is small, one can develop a perturbation expansion in the reactive part of the Liouville operator. This operator generates transitions upon collision when a given criterion on the line-of-centers kinetic energy is met. Such a theory holds for condensed systems and accounts for the environment of a colliding pair of particles by taking the transition probability to depend on particles near a given colliding pair. A major difference between this and the present study is that the end-product of these reactive events is not an aggregate, but rather a pair of particles, one or both of which have altered identity due to the reactive collision. The hypothesis on which the present study is based is that one can formulate an analogous multiscale approach for an N-atom system evolving under a continuous N-atom potential, and not by a hard-sphere model with a reactive transition probability. The key to our approach in this regard is that one can identify order parameters that are slowly varying in time. The order parameters we introduce characterize the coherent dynamics of the self-assembling nanocomponents.
Simulations of the dynamics of self-assembly involving molecular or nanoscale components have been performed using molecular dynamics (MD)29-31. Additionally, “lumped” or “coarsegrained” methods32,33 have been utilized to simulate the behavior of these and other biomolecular systems. The objective of these methods is to introduce a reduced set of variables (e.g. lumped clusters of atoms) and use heuristic arguments or calibration with experimental data to fit the interaction between these lumped elements. We wish to distinguish between a fully coded multiscale approach as adopted here (in Sect. III) and these other simulation methods. Consider the feedback loop of Fig.1. Here it is suggested that nanoscale features of the system (described via order parameters) can affect the probability distribution for atomistic fluctuations. In turn, these fluctuations create the entropy and the average forces that drive order parameter evolution.
This suggests that a computational approach to self-assembly should co-evolve the order parameters and the average forces acting on them. In contrast, coarse-grained or lumped methods provide an algorithm for computing forces on aggregates of atoms without accounting for the instantaneous value of the order parameters, thus ignoring the feedback loop of Fig.1. In addition to accounting for this interaction, the present multiscale approach provides a guideline for choosing the correct set of order parameters (e.g., the size of the aggregate of atoms constituting the lumped element).
The multiscale theory presented here is strongly based on an intuition regarding variables which characterize the long-time behavior of the system, i.e., over times much greater than those of atomic collisions and vibrations. These are generally collective variables, which represent the coherent dynamics of many atoms simultaneously. Multiscale theory enables one to capture the interplay of the coherent, many-atom and chaotic, individual-atom dynamics. The intuitive starting point of the theory not withstanding, our development provides a self-assembling test, notably that certain correlation functions have long-time tails34-36. If these are present they provide an indication that the proposed list of order parameters is not complete, i.e., there are other order parameters that couple to them strongly.
Depending on the importance of coherent inertial dynamics and friction effects, the result of multiscale theory is an equation of either FP or Smoluchowski type for the stochastic dynamics of the order parameters. We have developed the following six step procedure for the analysis of
multiscale systems23,24,26,27 :
Step 1: The system is described in terms of classical atoms interacting via a potential. Order
Newton’s equations and statistical arguments are used to show that the order parameters evolve on timescales that are long relative to that of atomic vibrations or
Step 4: An expansion of in powers of is introduced and the Liouville equation in its multiscale reformulation is solved order-by-order.
Step 5: The lowest order solution is assumed to reflect the near-equilibrium conditions relevant for many self-assembly problems, since the system has come to a steady state for the atomistic variables. Hence, this solution is taken to be independent of the time variable, which is designed to capture atomistic fluctuations. The Liouville equation implies
no further information is known about the lowest order distribution, an entropymaximization principle is used resulting in the construction of a set of possible distributions, each of which are applicable under distinct experimental conditions28.
Step 6: The solution to the Liouville equation at various orders in is examined. By asserting that the n-th order solution is well-behaved for large time and upon deriving a conservation law for the time evolution of the reduced probability density (depending
Smoluchowski equation is obtained. In this derivation, we do not ensure solvability conditions by integrating out the atomistic variables (e.g. the direct dependence of on ). Such a traditional approach leads to ambiguities when one wishes to use an allatom description of the system, and notably of the nanoscale subsystems of interest here. Rather, we use the Gibbs hypothesis which states that “the long-time and ensemble averages are equal near equilibrium”.
In the next section, this six-step procedure is developed, in which is not slowly-varying, to arrive at a theory of the self-assembly of nanocomponents into a composite. Implications for the numerical simulation of self-assembly are explored as well, in Sect. III. Finally, conclusions are drawn from the analysis and simulations in Sect. IV.