Model for stretching elastic biopolymers which exhibit conformational transformations

We derive an expression that represents the physical behavior of a polysaccharide molecule as it is stretched from the entropic region, through one or more ring conformational transformations, into the Hookean regime. The model adapts existing models in order to accommodate one or more force-induced conformational transformations of the glycan rings and is based on the concept of equilibrium between the clicked (longer conformers) and unclicked states. This equilibrium is determined by the Gibbs energy difference between these two states which is perturbed in favor of the clicked states by the force applied to the molecule. The derived expression is used to generate force-extension curves for model polymers and can illustrate the effect of the Gibbs energy for each transformation on the shape of these curves. It is also used to fit the force-extension curves of polysaccharides to obtain the Gibbs energy differences between the conformers. Good agreement was found between this model and experimental data on carboxymethylamylose, dextran, alginate, and pectin.

Figure 1

Gibbs energy barrier $\Delta G^{‡}$ and energy difference between the unclicked (shorter) and clicked (longer) states, $\Delta G_0$.

Figure 2

Simulated force-extension curves of polysaccharides with zero, one, or two clicks. Parameters use are $l_p=1\mathrm{nm}$, $N_{\mathit{\text{total}}}=200$, $\Delta x_1=0.07$, $\Delta x_2=0.03$, $\Phi =10\mathrm{nN}$, $\Delta G_{0_1}=12.05\mathrm{kJ}{\mathrm{mol}}^{-1}$, $\Delta G_{0_2}=18.07\mathrm{kJ}{\mathrm{mol}}^{-1}$, and $\text{temperature}=298\mathrm{K}$.

Figure 3

Simulated force extension curves for a two click polysaccharide with varying $\Delta G_0$. Solid line: $\Delta G_{0_2}>\Delta G_{0_1}$. Dotted line: $\Delta G_{0_2}=\Delta G_{0_1}$. Dashed line: $\Delta G_{0_2}=0$. Dot-dashed line: $\Delta G_{0_2}=-\Delta G_{0_1}$.

Figure 4

Force-extension curves (a) of dextran in ${\mathrm{H}}_2\mathrm{O}$ and (b) of CMA in ${\mathrm{H}}_2\mathrm{O}$. Open circles are experimental data; the solid line is the fitted model.

Figure 5

Force extension curves of (a) pectin and (b) alginate (70% guluronic acid). Open circles are experimental data while the solid lines are the fitted model for two clicks. For clarity, arrows mark the midpoints of the first (1) and second (2) force-induced transformations as calculated by $\Delta G_0∕\Delta x$. The midpoint of these transformations for pectin as $450\mathrm{pN}$ and $1250\mathrm{pN}$ and for alginate $240\mathrm{pN}$ and $630\mathrm{pN}$.

Figure 6

Force extension curve of pectin fitted with the (a) no-click model, (b) one-click model, and (c) two-click model. Residuals are shown in the inset.