布朗馬達(Brownian motors)的概念可以用來解釋受到熱擾動作用的分子馬達如何行機械化學耦合運動達成方向性的運動。其機械化學耦合是利用ATP的水解和鍵結來改變分子馬達構形，造成位能的改變而產生機械運動。為了讓布朗馬達產生單方向運動，並且讓速度有效提升，本文採用Rocking ratchet和Diffusion ratchet耦合的運動模型來模擬布朗馬達的運動情況。

本文使用Fokker-Planck equation為運動方程式，求解出在週期外力作用與變溫情況下的或然率與機率流，再利用文獻[11] Robust 數值演算法計算位能切換過程中的運作模式。此演算法涵蓋精細平衡 (Detailed balance) ，可用來描述具有連續馬可夫過程 (Continuous Markov process) 之性質的跳躍過程(Jump process) 與跳躍率 (Jump rate)，進而求解布朗馬達的運動性質。

採用MATLAB 7軟體計算以上描述之方程式及數值演算法，並與文獻[7]之結果比較，確認此求解週期性外力作用方法為正確，並且討論平均速度與等效擴散係數的影響。本文假設布朗馬達在考慮受到溫度變化、週期性外力變化與週期性位能能障變化的模型，並探討各模型對布朗馬達運動之平均速度與等效擴散係數的影響。

The concept of Brownian motors may use for to explain how molecular motors take mechanochemistry coupling of the motion to achieve the directive movement when it subjects to thermal fluctuation. The conformation of molecular motors is changed by the mechanochemistry coupling of the motion with hydrolysis of ATP and the binding of ATP. It means that molecular motors generate the change of potential energy and mechanical motion. In order to Brownian motors produce motion in one direction, and increasing the average velocity effectively. This paper uses the coupling model of Rocking ratchet and Diffusion ratchet to simulate the movement situation of Brownian motors.

The Fokker-Planck equation is the governing equation in this paper, it can be used to solve the probability and the probability flux in different temperature conditions and it also used to calculate working model with potential energy switches between an on and an off state by Robust numerical algorithm in the reference[11]. In the algorithm a continuous Markov process is discretized as a jump process and the jump rates are derived from local solution of the continuous systems which contain the property of detailed balance.

This paper used MATLAB 7.0 to calculate the equations and the numerical algorithm to analyze what kinds of different temperature conditions influences the average velocities of Brownian motors to achieve the maximum value, and to discuss the average velocities, the effective diffusion coefficients of Brownian motors with increasing the amplitude of potential energy under external force.

[1] Astumian, R. D.and Bier, M., Mechanochemical Coupling of Molecular Motors to ATP Hydrolysis, Biophysical Journal, 68, 219, (1995)

[2] Astumian, R. D. and Bier, M., Mechanochemical Coupling of the Motion of Molecular Motors to ATP Hydrolysis, Biophysical Journal, 70, 637-653, (1996).

[3] Elston, T. C., A Marcoscopic Description of Biomolecular Transport, J. Math. Biol, 41, 189-206, (2000).

[4] Nishiyama, M., Higuchi, H. and Yanagida T., Chemomechanical coupling of the forward and backward steps of single kinesin molecules, Nature Cell Biol ,4, 790-797, (2002).

[5] Keller, D. and Bustamante C., The Mechanochemistry of Molecular Motors, Biophysical Jouranl, 78, 541-556, (2000).

[6] Ai, B. Q., Wang, X. J., Liu, G. T., Wen, D. H., Xie, H. Z., Chen, W. and Liu L. G., Efficiency optimization in a correlation ratchet with asymmetric unbiased fluctuations, Physical Review E, 68,061105, (2003).

[7] Ai, B. Q., Wang, L., and Liu, L. G., Transport reversal in a thermal ratchet, Physical Review E, 72, 031101, (2005).

[8] 賴冠旭, 生物分子馬達運動分析, 國立成功大學工程科學系碩士論文, (2006).

[9] 楊秉凡, 切換速率對布朗馬達運動分析, 國立成功大學工程科學系碩士論文, (2007).

[10] 陳相慶,布朗馬達於變溫下之運動分析,國立成功大學工程科學系碩士論文,(2007).

[11] Wang, H., Peskin, C. and Elston, T., A Robust numerical algorithm for studying biomolecular transport processes. Theor. Biol, 221, 491-511, (2003).

[12] Risken, H., The Fokker-Planck Equation, Springer-Verlag, Berlin, (1996).

[13] Astumian, R. D. and Bier, M., Fluction Driven Ratchets: Molecular Motors. Phys Rev Lett, 72, 1766–1769, (1994).

[14] Smoluchowski, M. V., Experimentell nachweisbare, derüblichen Thermodynamik widersprechende Molekularph¨anomene. Physik. Zeitschr, 13, 1069, (1912).

[15] Feynman, R. P., Leighton, R. B. and Sands, M., The Feynman Lectures on Physics, 1, chapter 46, Addison Wesley, Reading MA, (1963).

[16] Kelly, T. R., Tellitu, I., and Sestelo, J. P., In search of molecular ratchets. Angew.Chem. Int. Ed. Engl, 36, 1866-1868, (1997).

[17] Kelly, T. R., Tellitu, I. and Sestelo, J. P., New molecular devices: In search of a molecular ratchet, J. Org. Chem, 63, 3655-3665, (1998).

[18] Davis, A. P., Tilting at windmills? The second law survives, Angew, Chem. Int.Ed. Engl, 37, 909-910, (1998).

[19] Sebastian, K. L., Molecular ratchets: verification of the principle of detailed balance and the second law of dynamics, Phys. Rev. E, 61 937, (2000).

[20] Reimann, P., Brownian motors: noisy transport far from equilibrium, Phys. Rep., 361, 57-265, (2002).

[21] Peskin, C.S. and Oster, G. F., Force production by depolymerizing microtubules: load-velocity curves and. run-pause statistics. Biophys. J., 69, 2268-2276, (1995).

[22] Ai, B. Q., Wang, L. and Liu, L. G., Brownian micro-engines and refriger- ors in a spatially periodic temperature field: Heat flow and performances, Physics Letters A, 352, 286–290, (2006).