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The polymerization energy (Gibbs free energy change of the polymerization reaction) can be written as equation(22) gpoly=?kTln(ka[m]kd),such that negative gpoly favors the growth of polymers. On the other hand, polymer elongation is mechanically unfavorable, as it leads to an increase in E. If we define ��E(N)?= E(N) �C E(N �C 1), the system free-energy increment upon addition of a monomer is equation(23) ��G(N)=��E(N)+gpoly.��G(N)=��E(N)+gpoly.As BGB324 mw a result of the competition between mechanical and chemical energies, the polymer will stop growing when the system free-energy increment ��G ceases to be negative. In Fig.?4, we plot ��G(N) versus polymer length N. The value ��G increases monotonically for small N. The preferred polymer length occurs where the curve intercepts ��G?= 0, such that additional growth is unfavorable. Varying material parameters, we often find moderate preferred lengths with 20?Ramoplanin a constant value after the conformational transition introduced in the previous section. If the constant value is smaller than |gpoly|, the polymer will favor continuous growth. When the polymer http://www.selleckchem.com/products/dinaciclib-sch727965.html length is not divergent, the nucleation-limited mechanism favors longer filaments than the equilibrium mechanism does. The reason for this is that the rarity of nucleation events prevents the free monomers from forming many short filaments, even though this configuration is energetically more favorable than fewer, longer filaments. Our result that membrane mechanics combines with the chemical energy of polymerization to control polymer length may be tested experimentally by quantifying cytoskeletal filament length on a supported lipid membrane in?vitro (17). In such an experiment, one may control the?monomer chemical potential (concentration), define the membrane composition and pinning, and vary the membrane curvature by forming lipid membrane on substrates patterned with wells or grooves (44). We predict that polymer length will depend not only on the monomer concentration, but also on the mechanical parameters of the membrane and the geometry of the patterns. In various cytoskeletal systems, NTP-hydrolysis changes the chemical energy of polymerization and thus affects polymer length (45?and?46). Our model demonstrates that coupled membrane-polymer mechanics can serve as an alternative mechanism of polymer length control, in addition to NTP-hydrolysis dynamics.