本論文研究黏著叢集在時變外力下斷裂的過程。此叢集由 Nt 個平行排列的配體–受體的鍵結所組成，並在一個隨著時間線性增加的外力 F = Gamma t 作用下所有鍵結將會全部斷裂， Gamma 為叢集對外力的載荷率。我們選擇了兩種不同的單一配體–受體的結合率與分離率，並且利用蒙地卡羅模擬法來模擬叢集斷裂的過程。研究結果顯示黏著叢集的特性與一特徵力 fc ＝ Fc/Nt 及一特徵載荷率 Gamma c 有關。當 Gamma＜ Gamma c，叢集的斷裂發生在力為 fr ，其值接近但小於 fc 。在此 Gamma的範圍下，叢集的斷裂行為可比擬為一粒子在一維座標下跨越位能障礙的過程。在將此叢集系統的自由能 G (Nb, F) 近似為配體–受體鍵結數目 Nb 的三次多項式下，理論計算發現 < Fc - Fr > Nt^(-1/3) 正比於 [ln Gamma^(-1)]^(2/3)，此關係式亦在模擬結果中被證實。當 Gamma＝ Gamma c，任何叢集大小都在 fr 等於 fc 時發生斷裂。當 Gamma＞ Gamma c， fr 大於 fc 且 fr 隨載荷率快速地增大，我們亦發現擁有較多的總鍵結數 Nt 之叢集的反應速率方程式 ( rate equation ) 之數值解與模擬結果吻合。

This thesis studies the dissociation of an adhesion cluster under shared linear loading
theoretically. A cluster of ligand-receptor pairs in cell adhesion can be modeled as Nt
parallel weak bonds between two surfaces. The cluster dissociates under an applied force
F which increases linearly with time t, that is, F = Gamma t, where Gamma is the loading rate. Monte
Carlo simulations of master equation are performed with two choices of kon and koff ,
the rebinding and unbinding rates of a bond, respectively. Our simulations show that
there exist a critical force per bond fc = Fc/Nt and a critical loading rate Gamma c, and some
universal properties of the clusters are associated with these quantities. At Gamma < Gamma c, the
rupture force per bond fr is close to but lower than fc. In this regime, cluster dissociation
can be regard as a one-dimensional barrier crossing process. We approximate the free
energy of the adhesion cluster G(Nb, F) at given F by a cubic function of Nb, number
of closed bonds in the cluster. From analytical solutions we obtained a scaling relation
< Fc - Fr >Nt^(-1/3) ~ [ln Gamma^(-1)]^(2/3) + constant. This scaling relation is consistent with the
numerical simulations of the master equation. At Gamma = Gamma c, the cluster dissociation occurs
at fr = fc for any cluster size. At Gamma > Gamma c, fr > fc and fr increases rapidly with Gamma,
especially for small clusters. There is no free energy barrier when the clusters rupture
at fr > fc, the numerical solutions of rate equation agree with numerical simulations of
the master equation.

[1] G. M. Cooper, The Cell - A Molecular Approach, second edition. (Sinauer Asso-
ciates, Inc., Sunderland (MA), 2000).
[2] G. I. Bell, Science 200, 618 (1978).
[3] T. Erdmann and U. S. Schwarz, Phys. Rev. Lett. 92, 108102 (2004).
[4] C. C. Tang, Y. P. Chu, and H. Y. Chen, Phys. Rev. E 76, 061905 (2007).
[5] E.-L. Florin, V. T. Moy, and H. E. Gaub, Science 264, 415 (1994).
[6] R. Alon, S. Chen, K.D. Puri, E.B. Finger, T.A. Springer. J. Cell Biol. 138, 1169
(1997).
[7] R. Merkel, P. Nassoy, A. Leung, K. Ritchie, E. Evans. Nature 397, 50 (1999).
[8] K. Prechtel, A. R. Bausch, V. Marchi-Artzner, M. Kantlehner, H. Kessler, and R.
Merkel, Phys. Rev. Lett. 89, 028101 (2002).
[9] U. Seifert, Europhys. Lett. 58, 792 (2002).
[10] T. Erdmann and U. S. Schwarz, Europhys. Lett. 66, 603 (2004).
[11] E. Evans and K. Ritchie, Biophys. J. 72, 1541 (1997).
[12] O. K. Dudko, A. E. Filippov, J. Klafter, and M. Urbakh, Proc. Natl. Acad. Sci.
U.S.A. 100, 11378 (2003).
[13] H. A. Kramers, Physica (Amsterdam) 7, 284 (1940).
[14] R. Merkel, Phys. Rep. 346, 344 (2001).
[15] E. Evans, A. Leung, V. Heinrich, and C. Zhu, Proc. Natl. Acad. Sci. U.S.A. 101,
11281 (2004).
[16] D. T. Gillespie, J. Phys. Chem. 81, 2340 (1977).
[17] R. H. Landau and M. J. P¶aez, Computational Physics - Problem Solving with Com-
puters. (John Wiley and Sons, Inc., New York, 1997).
[18] N. G. van Kampen, Stochastic Processes in Physics and Chemistry, third edition.
(North Holland, Amsterdam, 1992).
[19] F. Li and D. Leckband, J. Chem. Phys. 125, 194702 (2006).