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研究生:賴政宇
論文名稱:利用珠鏈模擬長鏈分子在束縛狀態下的動態統計性質探討
論文名稱(外文):Statistics and dynamics of vibrated bead chains confined in thin channels – simulations to the confined polymers
指導教授:周亞謙
學位類別:碩士
校院名稱:國立清華大學
系所名稱:物理系
學門:自然科學學門
學類:物理學類
論文種類:學術論文
論文出版年:2013
畢業學年度:101
語文別:中文
論文頁數:45
中文關鍵詞:珠鏈生物模擬
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摘要
珠鍊在通道中作布朗運動的觀測過程,我們追蹤所定位的珠子確切位置並可確切分析內部擷取的資訊:珠鏈前後端點距離、利用前後端點距離所求得鬆弛時間和擴散係數等,珠鏈處於通道內的特性與文獻 [1][2][3][4]所提在Odijk 區間下的行為進行比較;對於前後端點平均距離<x>與前後端點距離的方均根σ成0.5次方關係,而針對文獻 [1]所定義前方常數的型式加以驗證,所得結果相干長度ξ_p 為2.4公分與實際量測值2.6公分接近;並且文獻 [1]中:所提出以王和高的觀點加以推導出自由能量的型式:F/(k_B T)=A/(L-x)-Bx,第一項與文獻 [3]提出處於束縛狀態下的自由能量型式相同;而第二項為分子會自然伸縮所造成的等效力量型式;進而能推導出前後端點距離的機率分布方程式;由於我們所使用珠鏈處於通道內的狀態與自由能型式兩項意義相符,所以得到前後端點距離的機率分布,能以文獻 [1]所推得前後端點距離的機率分布方程式進行擬合,並加以驗證方程式定義的常數型式:在機率分布方程式中,我們所得:常數A結果與原始長度L成1.95次方關係,常數B與通道大小Dav 成 -0.8次方關係;而平均前後端點距離對通道大小的關係 [3][8]提及為2/3次方關係,我們實驗所得結果成1次方關係;珠鍊的鬆弛時間 [3]:實驗結果鬆弛時間τ對於原始長度L會因處於不同通道大小下而產生改變,在通道寬度≤10.5公釐時,τ對L成0.8次方關係;在通道寬度>10.5公釐時,τ對L成2次方關係;而內部擷取為65顆τ對通道大小Dav 成2次方關係;擴散係數:內部擷取所得質心以MSD方法求得擴散係數Ddiffusion ,求得擴散係數與擷取珠鏈數目N大致成 -0.68次方關係,除了珠鏈處於5公釐下的擴散係數與擷取珠鏈數目N成 -0.4次方關係,可能由於在5公釐通道束縛下造成接近一維的運動導致;最後,量測珠鍊在通道內的黏滯係數ζ,由史托克斯-愛因斯坦關係式回推計算愛因斯坦關係式D_diffusion/(〖μk〗_B T) 所得常數C,我們所得常數C數值在不同通道下約在1~2之間;此結果符合愛因斯坦關係式。

Abstract
The positions of each individual bead of the chain are tracked during the random motion. The statistical and dynamical properties of vibrated bead chains confined in thin channels. In the Odijk regime, these properties are the same as those of the measured and/or expected for long chain polymers. A phenomenogical form of the free energy, which is different from the Flory-type free energy, of the bead chain can explain the measured distribution of the end-to-end distance. The diffusion constant and the relaxation time
In a confined environment, the conformation of a semi-flexible long chain polymer is elongated, and the dynamic properties of the polymer molecule, such as the diffusion constant and the relaxation time, are changed due to the interaction with the confining walls. Thus the behaviors of a confined polymer are much complicated compared to those of the freely relaxed polymers. However, the situation can be much simplified for the strongly confined regime (the Odijk regime), where at least one of the dimensions of the confining structure is smaller than the persistence length of the long chain polymer. There will be a transition region between the Odijk regime and the regime where the de Gennes regime, where the blob theory can be applied. In this study, we shall focus on the statistical and dynamical properties of polymer chain in the Odijk regime.
The Flory free energy can account for many statistical and dynamical properties of long chain polymers in unconfined space. There are two terms in the Flory free energy, one is the bending elastic energy, which is proportional to the square of end-to-end distance, the other is the repulsion energy due to the excluded-volume interaction between the monomers of the chain, which is inversely proportional to L. However, in the Odijk regime, the Flory-type free energy can not explain the average end-to-end distance as a function of the width of the confining channel, and the distribution function of the end-to-end distance (L). One obvious argument is that in the strongly confined situation (Odijk regime), the semi-flexible long chain polymer is highly extended, and the interaction between other monomers (except the nearest neighbors) is not possible. The only interaction is the random collision with the confining wall. Thus, the second term in the Flory-type free energy should be zero, and the applicability of the Flory-type free energy in the Odijk regime is questionable.
A new form free energy was recently proposed. Wang and Gao found that the extension of the strongly confined polymer is the same as the extension of an unconfined polymer by an effective force. Based on this finding, Su el. al. derived that the free energy can be written as
F(x)/kBT = A/(L-x) – B x,
where A and B depend on the contour length and the persistence length of the polymer, and on the size of the confining channel. The first term of this free energy is actually equal to the free energy used by Reisner et.al.. And the second term is an effective force to stretch the the confined polymer. The free energy in (1) was used to explain the extension and the distribution of the extension as a function of the channel width, and also the relaxation time, of the confined DNA molecule. In this work, we find the statistical and dynamical properties of the confined bead chains can also be accounted for with the above form of free energy.
Granular bead chain has long been a model system for the long chain molecules in many theoretical studies of the statistical and dynamical properties of the molecules. Recent experimental results(16,17) showed that the vertically vibrated granular bead chains have the properties of the long chain molecules, however, only in the projection to the two-dimensional (2D) space. In this study, the confined DNA molecule in random motion is simulated with a strongly vibrated long bead chain. The chain consists of 2.1 mm diameter hollow metal beads connected to each other with a loose link that sets the maximum distance between two adjacent beads to 2.8 mm. The mass of each individual bead is mb = 35 mg and the bead chain is semi-flexible with a persistence length of 28 mm (i.e., the length of 10 beads). The confining channels have rectangular cross sections of dimension D mm x 8 mm. The height of the channels is fixed to 8 mm so that the crossing of the chain will not occur. And the condition for the excluded volume effect will hold in all the experiments. The horizontal motion of the beads is influenced by the random collision with the walls of the channel and their nearest neighbors. Our experiments are carried out in channel with width D that varies from 3 mm to 120 mm. The length of the channels is fixed at 60 cm. The channels are mounted on a 60  60 cm2 vibration platform that oscillates sinusoidally in the vertical direction at 19.7 Hz. The vertical vibration is driven by a heavy-duty electromagnetic shaker. In the measurements of the mobility of the randomly moving bead chain, the platform is tilted slightly, so that the gravitational force can be used as the drive for the damped motion. The speeds during the downward slides are found to be constant terminal speed of the confined chains. A fast digital camera (frame rate up to 200 fps) is used to record the images of the moving bead chains. Then the positions of the individual beads are tracked for further analysis. For each set of measurement, more than 2 x 104 images are analyzed to get good statistics. We also measure the velocities of beads inside and outside the channel. From the statistics of the bead velocities, we obtain the effective granular temperature T which is defined as with v being the velocity of the beads.

目錄 1
第一章 序論 2
1-1 動機 2
1-2 介紹 3
第二章 實驗儀器與系統 11
2-1 EM-Shaker 11
2-2 訊號產生器 12
2-3 調控水平裝置 12
2-4 CCD裝置與影像擷取 13
2-5 分析系統 13
2-6 珠鏈特性 15
2-7 長形通道 16
第三章 實驗結果與分析 18
3-1珠鏈內部擷取資訊 (segment of beads chain ) 18
3-2鬆弛時間(relaxation time ) 30
3-3 擴散係數(diffusion constant ) 35
第四章 結論 42
參考文獻 44

參考文獻
[1] Su T, Das SK, Xiao M, Purohit PK (2011) Transition between Two Regimes Describing Internal Fluctuation of DNA in a Nanochannel. PLoS ONE 6(3): e16890. doi: 10.1371/ journal.pone.0016890
[2] Wang J, Gao H (2007) Stretching a stiff polymer in tube. J Mater Sci 42: 8838-8843
[3] Reisner W, Morton KJ, Riehn R, Wang YM, Yu Z, et al. (2005) Statics and dynamics of single DNA molecules confined in nanochannels. Phys Rev Lett 94: 196101
[4] Y. Wang, D. R. Tree and K. D. Dorfman, Simulation of DNA extension in nanochannel Macromolecules, 2011, 44, 6594–6604
[5] Jun S, Thirumalai D, Ha B (2008) Compression and stretching of a self-avoiding chain in cylindrical nanopores. Phys Rev Lett 101: 138101.
[6] Qian H, Sheetz MP, Elson EL. (1991) Single particle tracking. Analysis of diffusion and flow in two-dimensional systems. Biophys. J. 60: 910–21
[7] Saxton MJ., Jacobson K (1997) Single particle tracking: Application to membrane danymics Annu. Rev. Biophys. Biomol. Struct. 26, 373.
[8] Yang Y, Burkhardt T W., Gompper G (2007) Free energy and extension of a semiflexible polymer in cylindrical confining geometries. Phys Rev E 76, 011804

[9] Tree D R, Wang Y, Dorfman K D (2012) Mobility of a Semiflexible chain confined in a nanochannel. Phys. Rev. Lett 108: 228105.
[10] Moon S J, Swift J B, Swinney H L (2004) Steady-state velocity distributions of an oscillated granular gas. Phys. Rev E 69,011301
[11] Cifra P, Benkovа ́ Z, Bleha T (2009) Chain extension of DNA confined in channels.
J. Phys. Chem. B 113, 1843-1851
[12] Odijk T (2008) Scaling theory of DNA confined in nanochannels and nanoslits. (2008)
Phys. Rev. E 77, 060901
[13] Odijk T (1983) On the statistics and dynamics of confined or entangled stiff polymers. (1983) Macromolecules 16, 1340-1344
[14] Jung Y, Jun S, Ha B Y (2009) Self-avoiding polymer trapped inside a cylindrical pore: Flory free energy and unexpected dynamics. Phys. Rev. E 79, 061912
[15] Bonthuis D J, Meyer C, Stein D, Dekker C (2008) conformation and dynamics of DNA confined in slitlike nanofluidic channels. Phys. Rev. Lett. 101, 108303

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