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(a) Classical linear sequence model of living polymers. (b) Generalized model for living polymerization (with transformation energy ǫtr). (c) Double-filament model.
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Living polymers are formed by reversible association of primary units (unimers). Generally the chain statistical weight involves a factor σ < 1 suppressing short chains in comparison with free unimers. Living polymerization is a sharp thermodynamic transition for σ < 1 which is typically the case. We show that this sharpness has an important effect...
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... peptide tapes and fibrils [10,11]. a e-mail: nyrkova@cerbere.u-strasbg.fr b e-mail: semenov@ics.u-strasbg.fr Thermodynamics of equilibrium polymerization is es- sentially defined by the cooperativity parameter σ (see Eq. (14) in Sect. 3) which is proportional to the mean number of short living chains (dimers for the linear se- quence model of Fig. 1a) per one free unimer: σ ∼ c short X , where c short and X are equilibrium concentrations of short chains and free unimers, respectively. The parameter σ can be low for two main reasons ...
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... Transformation energy. Equilibrium polymerization often requires a transformation of a unimer from an inac- tive to an active state implying a high energy cost ǫ tr leading to σ ∼ e −ǫtr/kB T ≪ 1 (see Fig. 1b). The typical examples are: opening of S 8 rings in sulfur [3,2,13], coil- to-rod transition in oligo-peptides forming self-assembling tapes [12,10,[14][15][16], a conformational transition to an "assembly-competent" structure in flagellin [8] and in pro- teins forming amyloid fibrils ...
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... End-cap energy. [12] A unimer in the self-assembling structure may interact with more than 2 neighbors (this is true for cylindrical surfactant micelles [7] or for double-strand and multi-strand filament structures of pro- teins [8], see Fig. 1c). For such living polymers the energy of chain scission ǫ sc = 2ǫ cap (ǫ cap is the energy of one end-cap) can be significantly higher than the energy ǫ d re- quired to dissociate a single unimer from a long aggregate: ǫ sc > ǫ d . Concentration of free unimers is proportional to e −ǫ d /kB T , while concentration of short aggregates ...
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... the simplest example, let us consider a system of units (unimers) that can polymerize in a linear fashion form- ing reversible bonds (see Fig. 1a). The system is suffi- ciently dilute, so that excluded-volume interactions be- tween unimers/aggregates can be neglected. Then con- centration X of unassociated (free) unimers nearly equals their activity, X ≃ e µ/kB T , where µ is the unimer chem- ical potential, and k B T is the thermal energy considered as the energy unit below ...
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Citations
... There are several kinetic mechanisms with which living macromolecules can change their lengths: chain scission/recombination, end-interchange, bond-interchange. 8,9,45 The simplest and the most basic kinetic mechanism is known as end-growth/evaporation implying that living chains are mixed with unimers and that the chain structure evolves by attaching or loosing a terminal unimer. This kinetic mechanism is generally applicable to many living polymer systems including protein polymers (actin and microtubule filaments), 5, 10-14 supramolecular discotic structures (chromonics, dyes), 15,16 reversibly polymerizing systems 1,17,18 (living ionic or free radical 19,20 polymerization such as in poly(α-methylstyrene)), certain liquid sulfur, 21 and wormlike micelle 22 systems. ...
... This equation is in a qualitative agreement with our Eqs. (44) and (45) showing that the relaxation process involves two modes, fast and slow, with dramatically different relaxation times (t f = t 1 N −1 and t s = 4N 2 t 1 ). Moreover, the two analytical predictions coincide as long as the fast mode is concerned. ...
... This model is complementary to that previously considered (Ref. 45) where scissions/recombinations of any bonds were equally allowed. The number of polymer chains in the system can change in the course of the process. ...
End-growth/evaporation kinetics in living polymer systems with association-ready free unimers (no initiator) is considered theoretically. The study is focused on the systems with long chains (typical aggregation number N 1) at long times. A closed system of continuous equations is derived and is applied to study the kinetics of the chain length distribution (CLD) following a jump of a parameter (T-jump) inducing a change of the equilibrium mean chain length from N0 to N. The continuous approach is asymptotically exact for t t1, where t1 is the dimer dissociation time. It yields a number of essentially new analytical results concerning the CLD kinetics in some representative regimes. In particular, we obtained the asymptotically exact CLD response (for N 1) to a weak T-jump ( N0N - 1 1). For arbitrary T-jumps we found that the longest relaxation time t max 1 is always quadratic in N ( is the relaxation rate of the slowest normal mode). More precisely tmax4N2 for N 0 2N and tmaxNN0(1 - NN0) for N 0 2N. The mean chain length Nn is shown to change significantly during the intermediate slow relaxation stage t1 t tmax. We predict that Nn(t)-Nn(0)t in the intermediate regime for weak (or moderate) T-jumps. For a deep T-quench inducing strong increase of the equilibrium Nn (N N0 1), the mean chain length follows a similar law, Nn(t)t, while an opposite T-jump (inducing chain shortening, N0 N 1) leads to a power-law decrease of Nn: Nn(t)t-13. It is also shown that a living polymer system gets strongly polydisperse in the latter regime, the maximum polydispersity index r NwNn being r ≈ 0.77N 0N 1. The concentration of free unimers relaxes mainly during the fast process with the characteristic time tf ∼ t 1N0N2. A nonexponential CLD dominated by short chains develops as a result of the fast stage in the case of N0 1 and N 1. The obtained analytical results are supported, in part, by comparison with numerical results found both previously and in the present paper.
... Nucleated linear assembly has attracted quite a bit of attention too, albeit mostly in the context of the polymerization of proteins (Attri et al., 1991;Edelstein-Keshet and Ermentrout, 1998;Goldstein and Stryer, 1986;Hiragi et al., 1990;Lomakin et al., 1996;Oosawa, 1970;Oosawa and Asakura, 1975;Oosawa and Kasai, 1962;Powers and Powers, 2006;Sept and McCammon, 2001). Almost all theoretical studies available in this field focus on end association/evaporation kinetics, where single monomers attach and detach from the chains, but reversible scission/recombination kinetics, involving chain breakage and fusion, has also very recently been investigated (Nyrkova and Semenov, 2007). ...
... (There is no obvious upper limit to their size other than that set by the system size.) This makes the analysis non-trivial, notwithstanding that in some cases useful analytical results can be obtained, exact or approximate (Dubbeldam and van der Schoot, 2005;Nyrkova and Semenov, 2007;Oosawa and Kasai, 1962;O'Shaughnessy and Yu, 1995;Powers and Powers, 2006;van der Linden and Venema, 2007;Wentzel, 2006). Here, we do not wish to indulge in the often times highly technical kinetic analysis but instead focus on the prevalent physics at hand. ...
... The reason is that the model presumes infinite co-operativity (or activation) for the fraction assembled material, which then behaves as if the polymerization transition were an actual phase transition. More elaborate calculations for reversible scission/recombination kinetics show that if K a > 0: (i) the relaxation time t à does not actually diverge at X = 1 albeit that it can become very large because K a ( 1 and (ii) that t à is also sensitive to the precise value of K a (Nyrkova and Semenov, 2007). ...
A brief overview is given of coarse-grained models that describe co-operativity and nucleation in supramolecular polymers in solution and in the melt state. Co-operativity is necessary but not sufficient for nucleation, itself a requirement for the autosteric control of supramolecular polymerization.
Amphiphilic molecules in solution typically produce structures coming from cooperative interactions of many synergetically acting functional units. If all essential interactions are weak, the structure can be treated theoretically based on a free energy expansion for small interaction parameters. However, most self-assembling soft matter systems involve strong specific interactions of functional units leading to qualitatively new structures of highly soluble micellar or fibrillar aggregates. Here we focus on the systems with the so-called strongly interacting groups (SIGs) incorporated into unimer molecules and discuss the effects of packing frustrations and unimer chirality as well as the origins of spontaneous morphological chirality in the case of achiral unimers. We describe several theoretical approaches (overcoming the limitations of weak interaction models) including the concepts of super-strong segregation, geometrical mismatch and orientational frustration. We also review some recently developed phenomenological theories of surfactant membranes and multiscale hierarchical approaches based on all-atomic modeling of packing structures of amphiphilic molecules with SIGs. In particular, we discuss self-assembling structures in systems possessing simultaneously several distinct types of SIGs: solutions of beta-sheet oligopeptides (showing different fibrillar morphologies), aromatic diamide-ester molecules (forming membranes, helical ribbons and tubules), and triarylamine amide derivatives (producing light-controlled supramolecular nanowires).
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Although pathway-specific kinetic theories are fundamentally important to describe and understand reversible polymerisation kinetics, they come in principle at a cost of having a large number of system-specific parameters. Here, we construct a dynamical Landau theory to describe the kinetics of activated linear supramolecular self-assembly, which drastically reduces the number of parameters and still describes most of the interesting and generic behavior of the system in hand. This phenomenological approach hinges on the fact that if nucleated, the polymerisation transition resembles a phase transition. We are able to describe hysteresis, overshooting, undershooting and the existence of a lag time before polymerisation takes off, and pinpoint the conditions required for observing these types of phenomenon in the assembly and disassembly kinetics. We argue that the phenomenological kinetic parameter in our theory is a pathway controller, i.e., it controls the relative weights of the molecular pathways through which self-assembly takes place.
We outline the basic theoretical physical concepts used to describe statistical properties of polymer molecules. This chapter is focused on the equilibrium theory of polymer systems in dilute and concentrated regimes. Both mean-field theories and scaling approaches are introduced. We consider the flexibility mechanisms and chain conformations in ideal polymer systems (linear chains, living, and branched polymers), the excluded-volume interaction effects in good and poor solvents, scattering functions of polymers, and the long-range correlation and interaction phenomena in polymer melts.
Although pathway-specific kinetic theories are fundamentally important to describe and understand reversible polymerisation kinetics, they come in principle at a cost of having a large number of system-specific parameters. Here, we construct a dynamical Landau theory to describe the kinetics of activated linear supramolecular self-assembly, which drastically reduces the number of parameters and still describes most of the interesting and generic behavior of the system in hand. This phenomenological approach hinges on the fact that if nucleated, the polymerisation transition resembles a phase transition. We are able to describe hysteresis, overshooting, undershooting and the existence of a lag time before polymerisation takes off, and pinpoint the conditions required for observing these types of phenomenon in the assembly and disassembly kinetics. We argue that the phenomenological kinetic parameter in our theory is a pathway controller, i.e., it controls the relative weights of the molecular pathways through which self-assembly takes place.
