受限空间中单链高分子动力学行为的计算机模拟
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摘要
近年来,高分子链在受限环境中的动力学行为引起了人们极大的关注。对这一动力学行为的研究一方面有助于我们更好地理解细胞生物学中的很多现象；另一方面则因为其有着很广泛而又重要的科技应用前景。高分子链在受限环境中的动力学行为是一个极其复杂的过程,因而限制了很多实验手段的应用。另一方面,理论计算在处理高分子链单元的体积排斥效应时显得力不从心。由此,“计算机模拟”成为此领域中一种不可替代的重要研究手段。随着计算机硬件的发展和新的模拟技术和模型的提出,计算机模拟将会发挥更大的作用。
相比连续空间的全原子模型,粗粒化格子模型虽然忽略了部分结构细节,但是从计算机模拟速度看更为高效,能够以不太长的计算机机时获得想要的结果。格子模型特别适合于进行Monte Carlo模拟。通过选择恰当的算法,MonteCarlo模拟还能对体系的动力学过程进行研究。
本论文的主体部分采用格子空间的动态Monte Carlo模拟对单链高分子在受限空间中的动力学行为进行了研究。此外,为了便于与具有体积排斥作用(self-avoiding effect)的自规避行走(SAW)链所得到的结果进行比较,本文还对处在二维方形受限空间中的理想高分子链(高斯链)穿越纳米管道的动力学行为进行了理论研究。
本博士论文的主要创新性贡献可以分为以下几个部分：
1、理论研究了处在二维方形受限空间中的理想单链高分子穿越纳米管道进入另一尺寸较大的方形空间的动力学过程。计算了高分子链穿越过程中各状态下的自由能以及所需的穿越时间。链单元与管道之间的相互作用明显改变了穿越所需克服的势垒。管道长度增加时,穿越由受熵势垒控制转变为由链单元-管道相互作用控制。这种转变导致穿越时间随链长非单调变化。我们计算得到了穿越时间与高分子链长、管道长度以及链单元-管道相互作用的精确表达式。研究结果有助于人们理解各因素如何影响高分子链的穿越速度。
2、使用Monte Carlo方法模拟研究了二维方形受限空间中的单链高分子穿越狭窄管道的动力学过程。模拟得到了高分子链在穿越过程所需克服的自由能势垒大小及各步所需时间随链长N以及管道长度M的变化关系。熵势垒的大小线性依赖于链长及管道长度的变化；链长增加时,第一和第二步穿越所需时间τ1和τ2线性增加,而第三步所需时间τ3保持不变；如果保持链长不变,τ1-M2.62,τ2随M线性增加,τ3-M1.90。我们将扩散速度考虑进去,采用Fokker-Planck方程解析计算了穿越各步所需时间随N和M的依赖关系,所得结果与模拟结果很好地吻合。
3、在前面工作的基础上,进一步研究了链单元-管道相互作用能(?)对处在受限方形空间中的高分子链穿越纳米管道动力学行为的影响。模拟结果揭示,高分子链的整个穿越过程可分为两个阶段,即高分子链的受限阶段和成功穿越阶段,所需时间分别定义为τtra[以及τesc。log(τtrap)随链长N以及管道长度M的增加线性变化。对于前者,当链单元-管道相互作用能(?)增加时,线性关系的斜率保持不变；而对于后者,线性关系的斜率随着(?)的增加而变大。链单元-管道之间存在较强的排斥或吸附作用时,高分子链单元很难进入或离开管道,导致高分子链的穿越时间τesc。迅速增大。
4、采用理论计算和Monte Carlo模拟研究了单链高分子从一温度较低(T1)的半无限大二维空间穿过小孔进入另一温度较高(T2)的半无限大空间的动力学过程。两种方法所得结果吻合的很好。结果表明,当两侧空间存在温度差异时,正的空间化学势有利于高分子链从低温区域进入高温区域：高分子链穿越小孔所需时间τ与化学势△μ反比,而与链长N成正比；温差△T(=T2-T1)越大,丁越小。
5、采用“键涨落”模型(BFM),对电场作用下单链DNA分子穿越深度不同交替排列的纳米管道的动力学行为进行了模拟研究。三维空间中,随着电场强度(?)的增加,DNA链的迁移率逐渐增加并渐趋一饱和值；场强(?)相同时,长链DNA的迁移率大于短链分子；DNA链进入窄管道前越过熵势垒所经历的等待时间即熵受限时间τtrap指数依赖于外加场强的倒数。模拟结果与实验结果很好地吻合。对高分子链在熵受限阶段的构象变化的细致研究揭示了装备对不同长度的高分子链的分离机制。DNA链在接近宽窄管道交界处时在电场作用下沿界面方向伸展变形,与短链分子相比,长链DNA有更多的链单元有机会接触界面,从而提高了越过熵势垒的几率,带动整条链进入窄管道,因此熵受限时间较短,平均迁移率较大。对DNA链在二维管道中的穿越过程的模拟研究所得结果证明了我们对上述分离机制的分析的正确性。
6、将Monte Carlo方法和“键涨落”模型(BFM)相结合,模拟了受限于两平行平面间的单链DNA分子在外力作用下的伸展以及撤除外力后弛豫的动力学过程,研究了受限程度对DNA分子的伸展长度及弛豫过程的影响。模拟结果表明,随着受限程度的增加,DNA分子链的构象更加伸展,这主要是由于随着平面间距的减少,DNA分子不同链段之间流体力学相互作用将会被平面屏蔽所致,受限程度不同时DNA分子的弛豫过程进一步证实了这一点。DNA分子的伸展长度(即末端距)随着流速的变化关系与实验结果及其对此所做的理论分析是基本一致的。
7、在电场作用下,DNA链的一部分链段进入受限程度较高的低熵区域而其余链段仍滞留在高熵的自由区域。电场撤除后,进入受限区域的链段将会自发回弹到自由区域。我们采用Monte Carlo模拟研究了这一动力学过程。模拟结果揭示,DNA链的回弹速度由慢到快逐渐增加,这与文献给出的实验现象及建立的理论模型是一致的,DNA链的回弹应归因于链单元在两个区域不同受限程度所产生的熵力的推动。模拟还研究了形成熵受限区域的障碍物的尺寸及间距对DNA链回弹速度的影响,进一步的分析表明链单元在受限区域的熵正比于链单元所能到达的自由空间占整个受限区域的比例。
Studying the dynamics of a polymer chain in the confined medium has attracted considerable attention in recent years. Understanding this process is very helpful for us to comprehend many phenomena in cell biology. On the other hand, it has widespread and important prospect on technological applications. The dynamics of a polymer chain in the confined medium is complicated, thus resulting in much difficulty of di-rect experimental detection in detail. However, theoretical calculation faces some dif-ficulties in dealing with the excluded-volume effect. As an alternate, computer simu-lation has been an important and useful method in this field. With the development of computer hardware and the putting forward of new simulation technology and model, computer simulation will play a more important role.
Compared with all-atom model in continuous space, the coarse-grained lattice model is, at the cost of spatial resolution, beneficial for CPU time. Lattice model is specifically suitable for Monte Carlo simulation. Monte Carlo simulation can be applied to study the dynamics process, if a proper algorithm is adopted.
This Ph.D thesis focuses on the dynamics process of a single polymer chain in confined mediums with dynamical Monte Carlo simulation. To compare with the sim-ulation results of a SAW chain, we study the translocation process of an ideal chain confined in a square through a nano-scale channel theoretically.
The main achievements are summarized as follows:
1. Theoretically study the translocation dynamical process of a single Gaussian chain from a square to another larger square through a nano-scale channel. The free energy barrier and mean translocation timeτare calculated. The potential interaction between the polymer and channel significantly modifies the entropic barrier landscape of translocation. As the channel length increases, the translocation process undergoes a transition from entropic barrier mechanism to a mechanism dominated by the pore-polymer interaction. This shift in mechanism leads to nonmonotonic dependence ofτon the pore length. Explicit formulas are derived for the dependence ofτon chain length, pore length, sizes of the donor and recipient squares. The calculated results provide guidance for tuning the rate of polymer translocation through narrow pores.
2. Study the dynamical process of the translocation for a single SAW chain con- fined in a square through a narrow channel with Monte Carlo simulation. The de-pendence of the height of the entropic barrier that the polymer need overcome when entering the channel and translocation time on chain length N and pore length M are achieved. The height of the entropic barrier depends linearly on N and M. The translo-cation times for the first and second steps increase linearly with N and the third step time dos not vary when M keeps constant. On the other hand, the simulation results reveal thatτ1-M2.62,τ2 increases linearly with M andτ3-M1.90. Furthermore, We get the dependence of the translocation times for each step by solving the Fokker-Planck equation numerically and, the results are quite consistent with our simulation results.
3. The effct of the pore-polymer interaction on the translocation of a single poly-mer chain confined in a finite size square through an interacting nanopore to a large space has been studied using Monte Carlo methods. The whole process consists of two stages. In the first stage, the chain experiences a period of trapping time, Ttrap, to overcome the free energy barrier. In the second stage, the chain successfully es-capes through the nanopore without totally pulled back to the donor square, and the consumed time is defined as Tesc. log(τtrap) decreases linearly with the chain length N and, the slopes of the lines for different pore-polymer interaction (?) are same all. How-ever, log(τtrap) increases linearly with the length of the nanopore M and the slope of line increases with (?). It is found that strong attractive as well as repulsive pore-polymer interaction adds the difficulty of the chain translocation through the nanopore, leading to the nonmonotonical dependence of the translocation time on the pore-polymer in-teraction.
4. The translocation of a polymer chain through a narrow pore in a membrane induced by the temperature difference on two sides of the membrane is studied by using theoretical approach and Monte Carlo simulations. We investigate the dependence of the translocation timeτon the chemical potential per monomerμ, the polymer length N, and the temperature difference of two sides. Our results reveal that positive chemical potential per monomerμis favorable for the translocation of the polymer from cis (of T1) to trans (of T2) side for the case of Ti< T2. The translocation timeτis inversely proportional toμ, and increases linearly with N. IncreasingΔT will accelerate the translocation of the chain from cis to trans side.
5. Using bond-fluctuation model (BFM), study the the motion of a long charged polymer chain inside an array of microscopic entropic traps. In the three-dimensional space, with the electric field increasing, the mobility of the chain inside the arrays in-creases gradually to a constant value. For same fields, long polymers are faster than shorter ones. The mean trapping time for the molecules inside the entropic trap ar-ray, Ttrap, exponentially increases with the reciprocal of the electric field,1/∈. The simulation results are in good agreement with the experimental observations. The size-separation mechanism relies mainly on the overall deformation of the molecules as the chain approaches the narrow constrictions. The simulation study of the motion for a polymer chain inside the two-dimensional arrays validates our analysis for the separa-tion mechanism mentioned above.
6. Stretching and relaxation of a single DNA molecule tethered in a specially designed thin slit were studied using Monte Carlo simulation combined with 3D bond fluctuation model (BFM). It was found that the extension and relaxation of the single DNA molecule are greatly affected by the confined environment. If the extent of the confined environment is increased by decreasing the distance between the two planar surfaces of the slit, the extension of the single DNA molecule increases, due to the screening of the hydrodynamic interaction of DNA segments by the planar surfaces of the slit. The relaxation of the single DNA molecule in different confined environments verifies this assumption completely. The correlation between the end-to-end separation and flow velocity obtained by Monte Carlo simulation is in good agreement with either the experimental results or theoretical consideration reported previously.
7. A part of a long DNA chain was driven into the confined environment by an electric field, with the other remains in the higher-entropy region. Upon removal of the field, the chain recoils to the higher-entropy rcgion spontaneously. This dynamical process has been investigated by Monte Carlo simulations. The simulation reproduces the experimental observed phenomenon that the recoil of the DNA chain is initially slow and gradually increases in speed. It is due to a confinement-entropic force, distinct from entropic elasticity. How the dimension and spacing of the nanopillars influence the recoil velocity was investigated and further analysis suggests that the characteristic entropy per monomer in the confinement is proportional to the area fraction of free part in the confinement.
相比连续空间的全原子模型,粗粒化格子模型虽然忽略了部分结构细节,但是从计算机模拟速度看更为高效,能够以不太长的计算机机时获得想要的结果。格子模型特别适合于进行Monte Carlo模拟。通过选择恰当的算法,MonteCarlo模拟还能对体系的动力学过程进行研究。
本论文的主体部分采用格子空间的动态Monte Carlo模拟对单链高分子在受限空间中的动力学行为进行了研究。此外,为了便于与具有体积排斥作用(self-avoiding effect)的自规避行走(SAW)链所得到的结果进行比较,本文还对处在二维方形受限空间中的理想高分子链(高斯链)穿越纳米管道的动力学行为进行了理论研究。
本博士论文的主要创新性贡献可以分为以下几个部分：
1、理论研究了处在二维方形受限空间中的理想单链高分子穿越纳米管道进入另一尺寸较大的方形空间的动力学过程。计算了高分子链穿越过程中各状态下的自由能以及所需的穿越时间。链单元与管道之间的相互作用明显改变了穿越所需克服的势垒。管道长度增加时,穿越由受熵势垒控制转变为由链单元-管道相互作用控制。这种转变导致穿越时间随链长非单调变化。我们计算得到了穿越时间与高分子链长、管道长度以及链单元-管道相互作用的精确表达式。研究结果有助于人们理解各因素如何影响高分子链的穿越速度。
2、使用Monte Carlo方法模拟研究了二维方形受限空间中的单链高分子穿越狭窄管道的动力学过程。模拟得到了高分子链在穿越过程所需克服的自由能势垒大小及各步所需时间随链长N以及管道长度M的变化关系。熵势垒的大小线性依赖于链长及管道长度的变化；链长增加时,第一和第二步穿越所需时间τ1和τ2线性增加,而第三步所需时间τ3保持不变；如果保持链长不变,τ1-M2.62,τ2随M线性增加,τ3-M1.90。我们将扩散速度考虑进去,采用Fokker-Planck方程解析计算了穿越各步所需时间随N和M的依赖关系,所得结果与模拟结果很好地吻合。
3、在前面工作的基础上,进一步研究了链单元-管道相互作用能(?)对处在受限方形空间中的高分子链穿越纳米管道动力学行为的影响。模拟结果揭示,高分子链的整个穿越过程可分为两个阶段,即高分子链的受限阶段和成功穿越阶段,所需时间分别定义为τtra[以及τesc。log(τtrap)随链长N以及管道长度M的增加线性变化。对于前者,当链单元-管道相互作用能(?)增加时,线性关系的斜率保持不变；而对于后者,线性关系的斜率随着(?)的增加而变大。链单元-管道之间存在较强的排斥或吸附作用时,高分子链单元很难进入或离开管道,导致高分子链的穿越时间τesc。迅速增大。
4、采用理论计算和Monte Carlo模拟研究了单链高分子从一温度较低(T1)的半无限大二维空间穿过小孔进入另一温度较高(T2)的半无限大空间的动力学过程。两种方法所得结果吻合的很好。结果表明,当两侧空间存在温度差异时,正的空间化学势有利于高分子链从低温区域进入高温区域：高分子链穿越小孔所需时间τ与化学势△μ反比,而与链长N成正比；温差△T(=T2-T1)越大,丁越小。
5、采用“键涨落”模型(BFM),对电场作用下单链DNA分子穿越深度不同交替排列的纳米管道的动力学行为进行了模拟研究。三维空间中,随着电场强度(?)的增加,DNA链的迁移率逐渐增加并渐趋一饱和值；场强(?)相同时,长链DNA的迁移率大于短链分子；DNA链进入窄管道前越过熵势垒所经历的等待时间即熵受限时间τtrap指数依赖于外加场强的倒数。模拟结果与实验结果很好地吻合。对高分子链在熵受限阶段的构象变化的细致研究揭示了装备对不同长度的高分子链的分离机制。DNA链在接近宽窄管道交界处时在电场作用下沿界面方向伸展变形,与短链分子相比,长链DNA有更多的链单元有机会接触界面,从而提高了越过熵势垒的几率,带动整条链进入窄管道,因此熵受限时间较短,平均迁移率较大。对DNA链在二维管道中的穿越过程的模拟研究所得结果证明了我们对上述分离机制的分析的正确性。
6、将Monte Carlo方法和“键涨落”模型(BFM)相结合,模拟了受限于两平行平面间的单链DNA分子在外力作用下的伸展以及撤除外力后弛豫的动力学过程,研究了受限程度对DNA分子的伸展长度及弛豫过程的影响。模拟结果表明,随着受限程度的增加,DNA分子链的构象更加伸展,这主要是由于随着平面间距的减少,DNA分子不同链段之间流体力学相互作用将会被平面屏蔽所致,受限程度不同时DNA分子的弛豫过程进一步证实了这一点。DNA分子的伸展长度(即末端距)随着流速的变化关系与实验结果及其对此所做的理论分析是基本一致的。
7、在电场作用下,DNA链的一部分链段进入受限程度较高的低熵区域而其余链段仍滞留在高熵的自由区域。电场撤除后,进入受限区域的链段将会自发回弹到自由区域。我们采用Monte Carlo模拟研究了这一动力学过程。模拟结果揭示,DNA链的回弹速度由慢到快逐渐增加,这与文献给出的实验现象及建立的理论模型是一致的,DNA链的回弹应归因于链单元在两个区域不同受限程度所产生的熵力的推动。模拟还研究了形成熵受限区域的障碍物的尺寸及间距对DNA链回弹速度的影响,进一步的分析表明链单元在受限区域的熵正比于链单元所能到达的自由空间占整个受限区域的比例。
Studying the dynamics of a polymer chain in the confined medium has attracted considerable attention in recent years. Understanding this process is very helpful for us to comprehend many phenomena in cell biology. On the other hand, it has widespread and important prospect on technological applications. The dynamics of a polymer chain in the confined medium is complicated, thus resulting in much difficulty of di-rect experimental detection in detail. However, theoretical calculation faces some dif-ficulties in dealing with the excluded-volume effect. As an alternate, computer simu-lation has been an important and useful method in this field. With the development of computer hardware and the putting forward of new simulation technology and model, computer simulation will play a more important role.
Compared with all-atom model in continuous space, the coarse-grained lattice model is, at the cost of spatial resolution, beneficial for CPU time. Lattice model is specifically suitable for Monte Carlo simulation. Monte Carlo simulation can be applied to study the dynamics process, if a proper algorithm is adopted.
This Ph.D thesis focuses on the dynamics process of a single polymer chain in confined mediums with dynamical Monte Carlo simulation. To compare with the sim-ulation results of a SAW chain, we study the translocation process of an ideal chain confined in a square through a nano-scale channel theoretically.
The main achievements are summarized as follows:
1. Theoretically study the translocation dynamical process of a single Gaussian chain from a square to another larger square through a nano-scale channel. The free energy barrier and mean translocation timeτare calculated. The potential interaction between the polymer and channel significantly modifies the entropic barrier landscape of translocation. As the channel length increases, the translocation process undergoes a transition from entropic barrier mechanism to a mechanism dominated by the pore-polymer interaction. This shift in mechanism leads to nonmonotonic dependence ofτon the pore length. Explicit formulas are derived for the dependence ofτon chain length, pore length, sizes of the donor and recipient squares. The calculated results provide guidance for tuning the rate of polymer translocation through narrow pores.
2. Study the dynamical process of the translocation for a single SAW chain con- fined in a square through a narrow channel with Monte Carlo simulation. The de-pendence of the height of the entropic barrier that the polymer need overcome when entering the channel and translocation time on chain length N and pore length M are achieved. The height of the entropic barrier depends linearly on N and M. The translo-cation times for the first and second steps increase linearly with N and the third step time dos not vary when M keeps constant. On the other hand, the simulation results reveal thatτ1-M2.62,τ2 increases linearly with M andτ3-M1.90. Furthermore, We get the dependence of the translocation times for each step by solving the Fokker-Planck equation numerically and, the results are quite consistent with our simulation results.
3. The effct of the pore-polymer interaction on the translocation of a single poly-mer chain confined in a finite size square through an interacting nanopore to a large space has been studied using Monte Carlo methods. The whole process consists of two stages. In the first stage, the chain experiences a period of trapping time, Ttrap, to overcome the free energy barrier. In the second stage, the chain successfully es-capes through the nanopore without totally pulled back to the donor square, and the consumed time is defined as Tesc. log(τtrap) decreases linearly with the chain length N and, the slopes of the lines for different pore-polymer interaction (?) are same all. How-ever, log(τtrap) increases linearly with the length of the nanopore M and the slope of line increases with (?). It is found that strong attractive as well as repulsive pore-polymer interaction adds the difficulty of the chain translocation through the nanopore, leading to the nonmonotonical dependence of the translocation time on the pore-polymer in-teraction.
4. The translocation of a polymer chain through a narrow pore in a membrane induced by the temperature difference on two sides of the membrane is studied by using theoretical approach and Monte Carlo simulations. We investigate the dependence of the translocation timeτon the chemical potential per monomerμ, the polymer length N, and the temperature difference of two sides. Our results reveal that positive chemical potential per monomerμis favorable for the translocation of the polymer from cis (of T1) to trans (of T2) side for the case of Ti< T2. The translocation timeτis inversely proportional toμ, and increases linearly with N. IncreasingΔT will accelerate the translocation of the chain from cis to trans side.
5. Using bond-fluctuation model (BFM), study the the motion of a long charged polymer chain inside an array of microscopic entropic traps. In the three-dimensional space, with the electric field increasing, the mobility of the chain inside the arrays in-creases gradually to a constant value. For same fields, long polymers are faster than shorter ones. The mean trapping time for the molecules inside the entropic trap ar-ray, Ttrap, exponentially increases with the reciprocal of the electric field,1/∈. The simulation results are in good agreement with the experimental observations. The size-separation mechanism relies mainly on the overall deformation of the molecules as the chain approaches the narrow constrictions. The simulation study of the motion for a polymer chain inside the two-dimensional arrays validates our analysis for the separa-tion mechanism mentioned above.
6. Stretching and relaxation of a single DNA molecule tethered in a specially designed thin slit were studied using Monte Carlo simulation combined with 3D bond fluctuation model (BFM). It was found that the extension and relaxation of the single DNA molecule are greatly affected by the confined environment. If the extent of the confined environment is increased by decreasing the distance between the two planar surfaces of the slit, the extension of the single DNA molecule increases, due to the screening of the hydrodynamic interaction of DNA segments by the planar surfaces of the slit. The relaxation of the single DNA molecule in different confined environments verifies this assumption completely. The correlation between the end-to-end separation and flow velocity obtained by Monte Carlo simulation is in good agreement with either the experimental results or theoretical consideration reported previously.
7. A part of a long DNA chain was driven into the confined environment by an electric field, with the other remains in the higher-entropy region. Upon removal of the field, the chain recoils to the higher-entropy rcgion spontaneously. This dynamical process has been investigated by Monte Carlo simulations. The simulation reproduces the experimental observed phenomenon that the recoil of the DNA chain is initially slow and gradually increases in speed. It is due to a confinement-entropic force, distinct from entropic elasticity. How the dimension and spacing of the nanopillars influence the recoil velocity was investigated and further analysis suggests that the characteristic entropy per monomer in the confinement is proportional to the area fraction of free part in the confinement.
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