Session Z46: Invited Session: Self-Assembly of Proteins: From Capsids to Crystals

Theoretical approach to crystallization: foundations and application to proteins

A fundamental issue in the modern study of phase transitions is the description of the process of nucleation, i.e. the choices of nucleation pathways. Proteins, in particular, are well-known to sometimes crystallize by passing through a meta-stable amorphous state and simulation and theory have shown that this is also true of many other systems. The issue also arises in the important case of polymorphic materials. In all cases, the goal is to understand which pathway is favored and how this is affected by the external control parameters. In this talk, I discuss a theoretical description of nucleation that allows for the direct determination of nucleation pathways and of their relative probability of realization that takes into account both thermodynamics and kinetic effects. It is based on a formulation of nucleation as a fundamentally non-equilibrium process and fully incorporates the effect of free-energy landscapes, determined e.g. via Density Functional Theory, in a consistent manner.   

Elasticity Theory of Viral Capsids

The continuum elasticity theory of icosahedral thin shells has been applied with success to shape transitions of the protein shell surrounding the viral genome, the capsid. The talk presents an extension of thin-shell elasticity theory that is applicable to aggregates of \textit{functional biomolecules } at length scales comparable to that of the component molecules themselves. Unlike classical elasticity theory, the stress and strain fields have a network of \textit{mathematical discontinuities} along the interfaces of the proteins, due to the conformational incompatibility of packing proteins as well as to conformational transitions of the proteins. The method is applied to the P-II to EI transition of the protein shell of the virus HK97 driven by hexon skewing. The combination of the intrinsic stresses of icosahedral shells and the conformational pre-stress turns the P-II state into a ``critical'' state whose shape is independent of the bending and Young's moduli.   

The Role of Multivalent Counterions in Protein Crystallization

In this talk, I will give an overview of our recent studies on the phase behavior of model globular proteins in solution in the presence of multivalent counterions. We have shown that negatively charged globular proteins at neutral pH in the presence of multivalent counterions undergo a ``reentrant condensation (RC)'' phase behavior [1,2], i.e. a phase-separated regime occurs in between two critical salt concentrations, c* $<$ c**, giving a metastable liquid-liquid phase separation (LLPS) [3]. This reentrant phase behavior corresponds to an effective charge inversion of proteins as confirmed by zeta-potential measurements and supported by Monte Carlo simulations [1,2]. Crystallization from the condensed regime follows different mechanisms. Near c*, crystals grow following a classic nucleation and growth mechanism; near c**, the crystallization follows a two-step mechanism, i.e, crystals growth follows a metastable LLPS [3,4]. Nucleation rate is faster from the protein-poor phase than that from the protein-rich phase, which cannot be explained by the recent theories. SAXS measurements demonstrate that protein clusters act as precursors for crystal growth, which reduce the energy barrier of nucleation [4]. X-ray diffraction analyses on the high quality single crystals provide direct evidence of the crystal structure and cation binding sites [3]. The bridging effect of the metal cations explains the cluster formation.\\[4pt] [1] Zhang, F.; et al. \textit{Phys. Rev. Lett.} \textbf{2008}, 101, 148101.\\[0pt] [2] Zhang, F.; et al. \textit{Proteins }\textbf{2010}, 78, 3450.\\[0pt] [3] Zhang, F.; et al.\textit{ J. Appl. Cryst.} \textbf{2011}, 44, 755.\\[0pt] [4] Zhang, F.; et al. In preparation.   

Self-assembly and Evolution from protein complexes to DNA nanostructures

The remarkable ability of biological matter to robustly self-assemble into well defined composite objects excites the imagination, suggesting that these processes could perhaps be emulated through the judicious design of synthetic building blocks. We use statistical mechanics to uncover the design rules for self-assembly into well defined three dimensional composite objects. In Nature, the rules for self-assembly emerge from an evolutionary process. We show how some patterns in protein complexes can be explained by their evolutionary origin [1]. We also introduce a coarse-grained rigid nucleotide model of DNA that reproduces the basic thermodynamics of short strands: duplex hybridization, single-stranded stacking and hairpin formation, and also captures the essential structural properties of DNA: the helical pitch, persistence length and torsional stiffness of double-stranded molecules, as well as the comparative flexibility of unstacked single strands [2]. We apply the model to calculate the detailed free-energy landscape of one full cycle of DNA ``tweezers,'' a simple machine driven by hybridization and strand displacement. We also study other nanomachines as well as processes such as force-induced melting, cruciform formation and the self-assembly of DNA tetrahedra.\\[4pt] [1] The self-assembly and evolution of homomeric protein complexes Gabriel Villar, et al., Phys. Rev. Lett. 102, 118106 (2009\\[0pt] [2] Structural and thermodynamic properties of a coarse-grained model of DNA, Thomas E. Ouldridge, Ard A. Louis, Jonathan P.K. Doye, J. Chem. Phys. 134 , 085101 (2011)