DOI 10.1140/epje/i2005-10011-1 Eur. Phys. J. E 17, 283–306 (2005) T HE EUROPEAN P HYSICAL JOURNAL E Shape transformations of protein-like copolymer globules A.R. Khokhlov 3 , A.N. Semenov 2, a , and A.V. Subbotin 1 1 Institute of Petrochemical Synthesis, Russian Academy of Sciences, Moscow 119991, Russia 2 Institut Charles Sadron, 6 rue Boussingault, 67083 Strasbourg Cedex, France 3 Physics Department, Moscow State University, Moscow 119992, Russia Received 19 January 2005 / Published online: 29 June 2005 – c  EDP Sciences / Societ` a Italiana di Fisica / Springer-Verlag 2005 Abstract. Shapes of globules formed by amphiphilic multi-block-copolymers in a selective solvent are considered theoretically. We focus on copolymers consisting mostly of insoluble H-units forming large core surrounded by a shell of soluble P-blocks. It is shown that the globule becomes non-spherical when the effective shell tension is low enough. The resultant shape depends on the shell bending energy: it is prolate if this energy is larger than the elastic energy of the core, and oblate in the opposite case. The central result is the prediction of the formation of a surface pattern of fingers accompanying or even preempting the shape transition mentioned above. We elucidate and discuss the following finger morphologies: 1) nearly spherical knob; 2) a necklace of spherical beads extending away from the surface; 3) mostly cylindrical fingers; 4) large thorn-like fingers. The first 3 morphologies develop at equilibrium as the shell area increases (or, equivalently, the shell tension decreases). Considering the relevant kinetical aspects we show that formation of fingers is a nucleation and growth process, and that the energy of their equilibrium nucleation is likely to be high. Therefore, the finger formation may be delayed, and may actually occur in the regime where the plain spherical surface is metastable. It is the last morphology (thorn-like fingers) that characterizes the metastable regimes when the finger formation is controlled by a high activation energy. The universal features of the above predictions inviting experimental tests are discussed. PACS. 61.25.Hq Macromolecular and polymer solutions; polymer melts; swelling 1 Introduction Copolymers with complex/irregular chemical sequence open a wide range of opportunities to design new func- tional polymer systems: there are practically unlimited possibilities to tune the eventual properties by varying the chemical sequence in such copolymers [1]. This is similar in spirit to the route taken by nature in the evolution of biological macromolecules (e.g., protein enzymes). Proteins often form relatively dense globules which (unlike homopolymer globules) do not precipitate because they are protected by a hydrophilic shell. The idea is to try and design synthetic copolymers forming self-assembling structures resembling native protein globules at least in some basic respects related to their functioning. This ap- proach is inspired by the fantastic efficiency of protein functional structures. It is clear that the analogous syn- thetic copolymers must be amphiphilic, containing both hydrophobic (H) and hydrophilic (polar, P) monomer units. The self-assembling properties of such copolymers must depend on their primary structure, i.e. on the se- quence of the H- and P-units [2] (the copolymer sequences a e-mail: semenov@cerbere.u-strasbg.fr involving just two types of units, H and P, are obviously much simpler than proteins containing many chemically different amino acid units). A promising strategy for designing such “protein-like” synthetic copolymers was proposed recently [1]. Computer simulations, experimental studies and analytical calcu- lations showed that “protein-like” copolymers can form globules whose stability can be enhanced by tuning the primary chemical sequence [1,3–5]. The functioning of protein globules is largely defined by their geometrical shape and surface structure (apart from their stability with respect to precipitation). It is known that natural proteins can form various shapes, from ball-like to strand-like. However, it is still unclear which factors define the geometrical morphology, the shape of copolymer globules. An important goal of a theoretical development in this area is the prediction of sequence-structure relationships elucidating the most essential parameters (both molecular and physico-chemical) controlling the aggregated copoly- mer structures. That sort of problem seems to be ex- tremely complicated in the general case. The aim of the present study is more modest: we try and identify certain