我們使用歐拉角（ Euler Angles ）推導出了一根具有等向性彎曲剛度及自發曲率但無自發扭曲的均勻彈性桿之形狀方程，並且研究了螺旋解在外力及外力矩下的彈性性質及其穩定性，以及要形成螺旋桿的邊界條件。我們推導出判斷螺旋桿穩定性的判據。我們發現外力和外力矩一階導數為零的地方，和穩定特徵值變號的地方相同，並且負值的外力矩傾向讓螺旋不穩定。當能量出現自我交岔的行為時，我們發現到一次性伸長跳躍現象應該發生在交岔點上，並且討論了一般實驗中一次性伸長跳躍現象發生於其他地方的原因。 我們利用布朗動力學研究了在有限溫度下具有自發曲率的二維半柔軟生物高分子的力學性質。在我們的模型中，當外力超過0.03時，力跟伸長將成線性關係，即外力將佔主導地位而熵效應較小。因此若要考慮熵效應，在我們的模型中，外力的範圍要小於0.03。 We derive the shape equations in terms of Euler angles for an uniform elastic rod with isotropic bending rigidity and spontaneous curvatures but free of spontaneous torsion, and study within this model the elasticity and stability of a helical filament under uniaxial force and torque. We study the boundary conditions required to form a helical rod. We derive the criterion to determine the stability of a helix. We find that the first derivative of force and torque always shares the same zeroes with one of the stability eigenvalues. We find that a negative torque tends to make a helix unstable. The energy-force curve is self-crossed when the rod undergoes a transition. Moreover, we find that the crossover point always gives the lowest energy under a given force, therefore, in principle, the transition should occur at the crossover point, and we give a discussion on the reason why it happens at other point in experiments. We use Browian dynamics to study the elasticity of a semiflexible biopolymer with spontaneous curvature under a uniaxial force at finite temperature. We find that in our model, when the applied force exceeds 0.03, the relationship between force and extension is linear, i.e. force dominates the elastics. Therefore, if one wants to consider the effects of entropy in our model, the applied force should be smaller than 0.03.
