适用于DNA电荷输运研究的紧束缚模型方法
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摘要
作为生命遗传信息的载体,DNA是一类非常重要的生命大分子,其对绝大多数生命体的发育和机体运作起指导作用。近年来,由于在基因损伤修复等许多基本生命过程中的关键作用和在纳米电子学中作为分子器件的重要应用前景,DNA分子中电荷输运问题已成为多个学科比如化学、物理、生物前沿领域中的研究热点。在研究初期围绕DNA分子是否具有导电能力这一问题研究者们进行了诸多争论,不同实验中DNA表现出导体、半导体、绝缘体甚至是超导体等相差悬殊的导电行为。经过十多年来的研究,尽管对具体机制仍存在争议,研究者们已经基本认同DNA分子具有输运电荷的能力,但其程度受DNA分子的序列、碱基堆叠的完整性、构型涨落、溶剂环境以及电极(或电荷给体受体)与碱基的耦合等因素的影响很大,其中某些因素对DNA导电性的影响是颠覆性的。这些因素由分子微观结构决定,在实验上不容易控制,这就迫切要求理论计算学家通过模拟计算从微观角度研究DNA分子中的电荷输运行为,解释电荷输运机制。
对DNA这类生命大分子体系的电子结构的计算一直是理论计算领域中一个具有挑战性的课题,主要是因为生命大分子体系所含原子数目太多且不具备周期性,利用常规量子化学从头算方法所需的计算条件当前计算机能力尚不能承受。在DNA电荷输运的计算模拟研究中,紧束缚模型方法因其简单明了、物理图像清晰等优点而被广泛采用。该方法把DNA分子抽象为一维格点体系,每个格点提供一个在位轨道,电子在紧邻格点轨道间能发生跃迁,其状态通过在位能和迁移积分描述。通过设定这两个参数的值,紧束缚模型方法可以计算DNA中电荷输运所涉及的电子状态、空穴状态或前线电子能带结构。
尽管在DNA的电荷输运研究中紧束缚模型方法被广泛应用,但对其方法本身的研究还很少,这导致模型中参数取值缺少标准,各种文献中所采用的参数不一致,严重影响了该方法的准确性,不利于DNA电荷输运的研究。针对这一问题,我们从最基本的理论公式出发,根据DNA的分子结构和电子结构逐级做近似,推导出参数计算公式,建立系统化的参数计算方案,完善适用于DNA分子的紧束缚模型方法,使其能够更加合理地应用于DNA电荷输运研究。我们提出以下参数计算方案：
第一,对孤立格点体系作量子化学计算以得到格点在位轨道；
第二,对格点及其紧邻格点组成的子体系作量子化学计算,得到该子体系的单电子哈密顿作为计算参数专用的有效DNA哈密顿,总体系中其他原子作为点电荷作用于该有效哈密顿或直接忽略；
第三,计算有效DNA哈密顿基于格点在位轨道的矩阵元,将矩阵元带入相应公式得到在位能和迁移积分。
根据所提出的参数计算方案,我们对DNA的紧束缚模型方法做了以下研究：
通过计算各种序列组合下空穴和电子的紧束缚模型参数,研究了DNA分子的一级结构(碱基序列)对参数的影响。计算结果表明,格点在位能主要取决于格点代表的碱基种类,四种碱基的空穴在位能的顺序是G 以DNA分子中碱基对间距离和扭转角为代表,研究了DNA分子的二级结构(构象)对紧束缚模型参数的影响。计算结果表明,格点间迁移积分受构象尤其是扭转角的影响非常大。当格点间距离取标准值而扭转角为0°时,GG格点间和AA格点间的迁移积分均能达到近0.8eV,而当扭转角取标准值36°时,两类迁移积分均不足0.1eV。当固定扭转角而增加格点间距离时,格点间迁移积分随着距离的增加呈指数函数形式衰减。计算过程中我们还考虑了紧邻格点极化作用对迁移积分的影响,当格点二体体系明显不对称时,格点间的相互极化差异较大,导致不能利用前线轨道的能级分裂来计算迁移积分。另外,我们还分别计算了从大量晶体数据统计分析得出的平均A-型构象和平均B-型构象中所有序列组合对应在位能和迁移积分,比较了其差异。
我们把计算出的参数应用于SSH模型,研究了DNA分子的势阱势垒结构对其中空穴极化子的影响。研究表明单阱结构中空穴以极化子的形式存在于GC碱基对组成的势阱中；当势阱跨度较窄时,空穴电荷在势垒上有一定分布；势阱宽度为3个GC碱基对时,极化子最稳定。在周期性阱垒结构中,势阱和势垒的宽度对极化子形状都有显著影响。通过拟合格点间迁移积分同格点间距的关系,我们得出DNA链polyA-polyT和polyG-polyC中电子-声子耦合因子,然后将其应用于SSH模型以研究这两种链中空穴极化子的静态性质及其在电场下的动态行为。计算结果表明polyG-polyC中的空穴极化子的定域程度、稳定性均强于polyA-polyT中极化子,但在电场作用下polyA-polyT中极化子的迁移率较高,利于长程电荷迁移现象的发生。
除了对DNA分子的紧束缚模型方法研究外,我们还对之前提出的利用点群对称性简化矩阵元计算的生成子方法进行了一定研究。按照初始基函数在对称操作下的变换关系,可将初始基函数空间划分为对称性恒定子空间。在此基础上我们利用群表示理论和投影算符的性质从理论上证明了生成子方法,并将其推广到点群的高维不可约表示。作为演示,我们把该方法应用于苯分子的紧束缚模型计算和Heisenberg半经验价键模型计算中。
本论文的创新点主要包括：
1.从最根本的量子力学方程出发,通过逐级近似,得到了紧束缚模型参数的计算公式,据此建立了系统化的参数计算方案,完善了适用于DNA分子的紧束缚模型方法。
2.根据所提出的参数计算方案,研究了DNA分子紧束缚模型中在位能和迁移积分随DNA分子的序列、构象的变化关系。
3.根据在位能计算结果构建DNA势阱势垒结构,并利用SSH模型研究了其中空穴极化子的性质；根据迁移积分同格点间距离的关系得出SSH模型中电子-声子耦合因子。
4.基于对称性恒定子空间的特性,利用群表示理论证明了之前提出的生成子方法,并将其推广到点群的高维不可约表示,使这一方法得以完善。
Deoxyribonucleic acid (DNA) is a kind of important biomacromolecule that contains the genetic information used in the development and functioning of most known living organisms. Because of the key role in many of basic life processes such as gene damage and repair, as well as the promising application in molecular electronics, charge transport in DNA molecules has caught considerable attention of chemists, physicists and biologists in recent years. Charge-transfer reactions and conductivity measurements show a large variety of possible electronic behavior, ranging from Anderson and band-gap insulators to semiconductors, conductors, and even induced superconductors. After ten years of research, although the specific mechanism still remains controversial, the basic ability of charge migration through the DNA has been universally recognized. However the conductivity of DNA is greatly affected by many factors, such as the base sequence, the integrity of base stacking, the fluctuations of conformation, solvent environment, the electrode (or the charge donor and acceptor) and so on. These factors, some of which give a subversive influence on the conductivity of DNA, are found directly related to the microstructure of DNA. Due to the difficulties in experiment to control this, theoretical efforts should be made to understand the electronic properties of DNA and to look for common mechanism to elucidate charge transport in DNA.
The electronic structure calculation of complicated biomacromolecule such as DNA has been a challenging task for routine quantum computation methods, because these molecules are very large in size, and without periodic conditions. The tight-binding model has been extensively adopted to investigate electronic structure of DNA because of its advantages, for example, simplicity. In this tight-binding model approach, the backbone of DNA is generally ignored because it makes little contribution to the charge transfer. Each base or base pair is set to be one lattice site in the model, and each site provide only one site orbital for electron locate at this site. Electrons can hop between the adjacent site orbitals, and the electronic structure can be described by two parameters:the on-site energy and the transfer integral between two adjacent sites.
Although this tight-binding model approach has been extensively used, researches on this methods itself are seldom reported, resulting in lack of standard values of the two parameters. Parameters used in various literatures are inconsistent, seriously affecting the accuracy of this method. In view of this problem, we derived the parameter formula from the first principles with gradual approximation and put forward systematic scheme for calculating the two parameters. The detailed scheme is given as follows:
1. Quantum chemical calculations are carried out for all the isolated lattice system, and the site orbitals are obtained.
2. Quantum chemical self-consistent field calculations are carried out for the subsystem which contains the isolated lattice and its adjacent lattices. Then the single-electron Hamiltonian for this subsystem, which is an effective Hamiltonian for calculating the parameters, is obtained. The other atoms in the total system can be taken as point charges or be ignored.
3. The matrix elements of the effective Hamiltonian based on the site orbital are calculated. Then the on-site energy and the transfer integral can be calculated from these matrix elements using the corresponding formula.
According to the proposed parameter calculation scheme, our researches on the tight-binding model method of DNA are as follows:
By calculating the hole and the electron tight binding parameters of various base sequences, the influence of primary structure (base sequence) on the parameters are investigated. The results show that the on-site energy depends mainly on the base type. The order of the hole-on-site energy among the four type bases is G< A< T< C, while the order of electron-on-site energy is opposite. The on-site energy level splits because of the polarization of the adjacent lattices. As for the hole, this polarization leads to a lower energy and the 3'end gives a stronger polarization than the 5'end. The on site energy of guanine base shows the most obvious polarization effect among the four bases. The value of transfer integral of ideal B-form DNA ranges from 0.02 to 0.12eV, which depends on the bases type too. These parameters can be used to construct the tight-binding model Hamiltonian of arbitrary sequences of DNA molecules, calculate the hole or the electron states, and study the influence of the base sequence of the DNA on its charge-transport behavior.
Dependence of the two tight binding model parameters on the secondary structure (double-helical conformation) of DNA such as rise and twist was also investigated. The results show that the transfer integral between adjacent sites critically depends on the double-helical conformation, especially the twist between the two adjacent base pairs. When the distance between the two adjacent sites takes the standard value (3.38A), the transfer integral of AA and GG with the twist angle of 0°is nearly 0.8eV, but with the twist angle 36°, both types of transfer integral is under 0.1 ev. When the value of twist angle is fixed, the transfer integral decays exponentially with the increase of distance between the two adjacent sites. The influence of the polarization of the adjacent site on the transfer integral was discussed. The result show that, when two-site subsystem used to construct the effective DNA Hamiltonian is obviously asymmetric, the polarization of the site is different from each other, which result in the disuse of the frontier orbital splitting to calculate the transfer integral. In addition, we also compared the on site energy and the transfer integral differences between the average A-DNA and the B-DNA crystal structure.
With the parameters obtained, we investigated properties of polarons in different quantum well-barrier potential structures of DNA using the tight-binding SSH model. In the case of single-well potential structures, the hole localizes at the GC base pair, forming a polaron, and the polaron is most stable when the quantum well contains three GC base pairs. In the other case, namely periodic well-barrier potential structure, the width of the quantum wells and the quantum barriers both show a significant effect on the polaron. The electron-phonon coupling factor in SSH model is obtained by calculating the relationship between the transfer integral and distance in poly(G)-poly(C) and poly(A)-poly(T) DNA molecules. Properties of hole polarons in these two molecules are investigated using SSH model. The results of model calculation illustrate that polaron in poly(A)-poly(T) has a larger width and is more delocalized than that in poly(G)-poly(C) DNA molecule. Polarons in both the two kind of DNA molecules move in the form of drifting when a electric field is applied in the direction of the DNA chain, and polaron in poly (A)-poly (T) has a higher drifting mobility.
In addition to the investigations on the tight binding model method of DNA, we also do some study on the generator method to simplify the calculation of elements of the Hamiltonian matrix. When a system under consideration has some symmetry, usually its Hamiltonian space can be parallel partitioned into a set of subspaces, which is invariant under symmetry operations. A general algorithm to construct the generator functions is proposed, and is extended to deal with system having high-dimensional irreducible representations. Furthermore, two model Hamiltonians is used to show how the generator method simplifies the calculation of Hamiltonian matrix elements.
The original contributions in this doctoral dissertation are as follows:
1. Computing formula for the parameters of tight binding model was obtained from the most fundamental equation of quantum mechanics with the gradual approximation, and a systematic parameter calculation scheme has been proposed.
2. With the obtained parameter calculation scheme, the influences of primary structure and secondary structure on the parameters are investigated. The result illustrate that the on site energy depends on the base type of the lattice, and can be affected by the sequence of DNA molecule. While the transfer integral depends mainly on the conformation of DNA molecule, and the influence of the sequence of DNA molecule can be ignored.
3. Two types of special quantum well-barrier potential structures of the DNA chains are constructed based on the calculated on site energy, properties of hole polaron in them are investigated using the SSH model. Furthermore, with the electron-phonon coupling factor obtained by fitting dependence of transfer integral on distance, static and dynamic properties of hole polarons in poly(G)-poly(C) and poly(A)-poly(T) DNA molecules are investigated.
4. With the concept that symmetry-invariant subspace, the previously proposed generator method is proved through group representation theory and properties of projection operator, and then generalized to the circumstance of high-dimension irreducible representations of point group.
对DNA这类生命大分子体系的电子结构的计算一直是理论计算领域中一个具有挑战性的课题,主要是因为生命大分子体系所含原子数目太多且不具备周期性,利用常规量子化学从头算方法所需的计算条件当前计算机能力尚不能承受。在DNA电荷输运的计算模拟研究中,紧束缚模型方法因其简单明了、物理图像清晰等优点而被广泛采用。该方法把DNA分子抽象为一维格点体系,每个格点提供一个在位轨道,电子在紧邻格点轨道间能发生跃迁,其状态通过在位能和迁移积分描述。通过设定这两个参数的值,紧束缚模型方法可以计算DNA中电荷输运所涉及的电子状态、空穴状态或前线电子能带结构。
尽管在DNA的电荷输运研究中紧束缚模型方法被广泛应用,但对其方法本身的研究还很少,这导致模型中参数取值缺少标准,各种文献中所采用的参数不一致,严重影响了该方法的准确性,不利于DNA电荷输运的研究。针对这一问题,我们从最基本的理论公式出发,根据DNA的分子结构和电子结构逐级做近似,推导出参数计算公式,建立系统化的参数计算方案,完善适用于DNA分子的紧束缚模型方法,使其能够更加合理地应用于DNA电荷输运研究。我们提出以下参数计算方案：
第一,对孤立格点体系作量子化学计算以得到格点在位轨道；
第二,对格点及其紧邻格点组成的子体系作量子化学计算,得到该子体系的单电子哈密顿作为计算参数专用的有效DNA哈密顿,总体系中其他原子作为点电荷作用于该有效哈密顿或直接忽略；
第三,计算有效DNA哈密顿基于格点在位轨道的矩阵元,将矩阵元带入相应公式得到在位能和迁移积分。
根据所提出的参数计算方案,我们对DNA的紧束缚模型方法做了以下研究：
通过计算各种序列组合下空穴和电子的紧束缚模型参数,研究了DNA分子的一级结构(碱基序列)对参数的影响。计算结果表明,格点在位能主要取决于格点代表的碱基种类,四种碱基的空穴在位能的顺序是G
我们把计算出的参数应用于SSH模型,研究了DNA分子的势阱势垒结构对其中空穴极化子的影响。研究表明单阱结构中空穴以极化子的形式存在于GC碱基对组成的势阱中；当势阱跨度较窄时,空穴电荷在势垒上有一定分布；势阱宽度为3个GC碱基对时,极化子最稳定。在周期性阱垒结构中,势阱和势垒的宽度对极化子形状都有显著影响。通过拟合格点间迁移积分同格点间距的关系,我们得出DNA链polyA-polyT和polyG-polyC中电子-声子耦合因子,然后将其应用于SSH模型以研究这两种链中空穴极化子的静态性质及其在电场下的动态行为。计算结果表明polyG-polyC中的空穴极化子的定域程度、稳定性均强于polyA-polyT中极化子,但在电场作用下polyA-polyT中极化子的迁移率较高,利于长程电荷迁移现象的发生。
除了对DNA分子的紧束缚模型方法研究外,我们还对之前提出的利用点群对称性简化矩阵元计算的生成子方法进行了一定研究。按照初始基函数在对称操作下的变换关系,可将初始基函数空间划分为对称性恒定子空间。在此基础上我们利用群表示理论和投影算符的性质从理论上证明了生成子方法,并将其推广到点群的高维不可约表示。作为演示,我们把该方法应用于苯分子的紧束缚模型计算和Heisenberg半经验价键模型计算中。
本论文的创新点主要包括：
1.从最根本的量子力学方程出发,通过逐级近似,得到了紧束缚模型参数的计算公式,据此建立了系统化的参数计算方案,完善了适用于DNA分子的紧束缚模型方法。
2.根据所提出的参数计算方案,研究了DNA分子紧束缚模型中在位能和迁移积分随DNA分子的序列、构象的变化关系。
3.根据在位能计算结果构建DNA势阱势垒结构,并利用SSH模型研究了其中空穴极化子的性质；根据迁移积分同格点间距离的关系得出SSH模型中电子-声子耦合因子。
4.基于对称性恒定子空间的特性,利用群表示理论证明了之前提出的生成子方法,并将其推广到点群的高维不可约表示,使这一方法得以完善。
Deoxyribonucleic acid (DNA) is a kind of important biomacromolecule that contains the genetic information used in the development and functioning of most known living organisms. Because of the key role in many of basic life processes such as gene damage and repair, as well as the promising application in molecular electronics, charge transport in DNA molecules has caught considerable attention of chemists, physicists and biologists in recent years. Charge-transfer reactions and conductivity measurements show a large variety of possible electronic behavior, ranging from Anderson and band-gap insulators to semiconductors, conductors, and even induced superconductors. After ten years of research, although the specific mechanism still remains controversial, the basic ability of charge migration through the DNA has been universally recognized. However the conductivity of DNA is greatly affected by many factors, such as the base sequence, the integrity of base stacking, the fluctuations of conformation, solvent environment, the electrode (or the charge donor and acceptor) and so on. These factors, some of which give a subversive influence on the conductivity of DNA, are found directly related to the microstructure of DNA. Due to the difficulties in experiment to control this, theoretical efforts should be made to understand the electronic properties of DNA and to look for common mechanism to elucidate charge transport in DNA.
The electronic structure calculation of complicated biomacromolecule such as DNA has been a challenging task for routine quantum computation methods, because these molecules are very large in size, and without periodic conditions. The tight-binding model has been extensively adopted to investigate electronic structure of DNA because of its advantages, for example, simplicity. In this tight-binding model approach, the backbone of DNA is generally ignored because it makes little contribution to the charge transfer. Each base or base pair is set to be one lattice site in the model, and each site provide only one site orbital for electron locate at this site. Electrons can hop between the adjacent site orbitals, and the electronic structure can be described by two parameters:the on-site energy and the transfer integral between two adjacent sites.
Although this tight-binding model approach has been extensively used, researches on this methods itself are seldom reported, resulting in lack of standard values of the two parameters. Parameters used in various literatures are inconsistent, seriously affecting the accuracy of this method. In view of this problem, we derived the parameter formula from the first principles with gradual approximation and put forward systematic scheme for calculating the two parameters. The detailed scheme is given as follows:
1. Quantum chemical calculations are carried out for all the isolated lattice system, and the site orbitals are obtained.
2. Quantum chemical self-consistent field calculations are carried out for the subsystem which contains the isolated lattice and its adjacent lattices. Then the single-electron Hamiltonian for this subsystem, which is an effective Hamiltonian for calculating the parameters, is obtained. The other atoms in the total system can be taken as point charges or be ignored.
3. The matrix elements of the effective Hamiltonian based on the site orbital are calculated. Then the on-site energy and the transfer integral can be calculated from these matrix elements using the corresponding formula.
According to the proposed parameter calculation scheme, our researches on the tight-binding model method of DNA are as follows:
By calculating the hole and the electron tight binding parameters of various base sequences, the influence of primary structure (base sequence) on the parameters are investigated. The results show that the on-site energy depends mainly on the base type. The order of the hole-on-site energy among the four type bases is G< A< T< C, while the order of electron-on-site energy is opposite. The on-site energy level splits because of the polarization of the adjacent lattices. As for the hole, this polarization leads to a lower energy and the 3'end gives a stronger polarization than the 5'end. The on site energy of guanine base shows the most obvious polarization effect among the four bases. The value of transfer integral of ideal B-form DNA ranges from 0.02 to 0.12eV, which depends on the bases type too. These parameters can be used to construct the tight-binding model Hamiltonian of arbitrary sequences of DNA molecules, calculate the hole or the electron states, and study the influence of the base sequence of the DNA on its charge-transport behavior.
Dependence of the two tight binding model parameters on the secondary structure (double-helical conformation) of DNA such as rise and twist was also investigated. The results show that the transfer integral between adjacent sites critically depends on the double-helical conformation, especially the twist between the two adjacent base pairs. When the distance between the two adjacent sites takes the standard value (3.38A), the transfer integral of AA and GG with the twist angle of 0°is nearly 0.8eV, but with the twist angle 36°, both types of transfer integral is under 0.1 ev. When the value of twist angle is fixed, the transfer integral decays exponentially with the increase of distance between the two adjacent sites. The influence of the polarization of the adjacent site on the transfer integral was discussed. The result show that, when two-site subsystem used to construct the effective DNA Hamiltonian is obviously asymmetric, the polarization of the site is different from each other, which result in the disuse of the frontier orbital splitting to calculate the transfer integral. In addition, we also compared the on site energy and the transfer integral differences between the average A-DNA and the B-DNA crystal structure.
With the parameters obtained, we investigated properties of polarons in different quantum well-barrier potential structures of DNA using the tight-binding SSH model. In the case of single-well potential structures, the hole localizes at the GC base pair, forming a polaron, and the polaron is most stable when the quantum well contains three GC base pairs. In the other case, namely periodic well-barrier potential structure, the width of the quantum wells and the quantum barriers both show a significant effect on the polaron. The electron-phonon coupling factor in SSH model is obtained by calculating the relationship between the transfer integral and distance in poly(G)-poly(C) and poly(A)-poly(T) DNA molecules. Properties of hole polarons in these two molecules are investigated using SSH model. The results of model calculation illustrate that polaron in poly(A)-poly(T) has a larger width and is more delocalized than that in poly(G)-poly(C) DNA molecule. Polarons in both the two kind of DNA molecules move in the form of drifting when a electric field is applied in the direction of the DNA chain, and polaron in poly (A)-poly (T) has a higher drifting mobility.
In addition to the investigations on the tight binding model method of DNA, we also do some study on the generator method to simplify the calculation of elements of the Hamiltonian matrix. When a system under consideration has some symmetry, usually its Hamiltonian space can be parallel partitioned into a set of subspaces, which is invariant under symmetry operations. A general algorithm to construct the generator functions is proposed, and is extended to deal with system having high-dimensional irreducible representations. Furthermore, two model Hamiltonians is used to show how the generator method simplifies the calculation of Hamiltonian matrix elements.
The original contributions in this doctoral dissertation are as follows:
1. Computing formula for the parameters of tight binding model was obtained from the most fundamental equation of quantum mechanics with the gradual approximation, and a systematic parameter calculation scheme has been proposed.
2. With the obtained parameter calculation scheme, the influences of primary structure and secondary structure on the parameters are investigated. The result illustrate that the on site energy depends on the base type of the lattice, and can be affected by the sequence of DNA molecule. While the transfer integral depends mainly on the conformation of DNA molecule, and the influence of the sequence of DNA molecule can be ignored.
3. Two types of special quantum well-barrier potential structures of the DNA chains are constructed based on the calculated on site energy, properties of hole polaron in them are investigated using the SSH model. Furthermore, with the electron-phonon coupling factor obtained by fitting dependence of transfer integral on distance, static and dynamic properties of hole polarons in poly(G)-poly(C) and poly(A)-poly(T) DNA molecules are investigated.
4. With the concept that symmetry-invariant subspace, the previously proposed generator method is proved through group representation theory and properties of projection operator, and then generalized to the circumstance of high-dimension irreducible representations of point group.
引文
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2 Hall D B, Holmlin R E, Barton J K. Oxidative DNA damage through long-range electron transfer. Nature,1996,382:731-735
3 Giese B. Long distance charge transport in DNA:The hopping mechanism. Acc Chem Res, 2000,33:631-636
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7 Takada T, Kawai K, Fujitsuka M, Majima T. Direct observation of hole transfer through double-helical DNA over 100 A. PNAS,2004,101:14002-14006
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2 Hall D B, Holmlin R E, Barton J K. Oxidative DNA damage through long-range electron transfer. Nature,1996,382:731-735
3 Giese B. Long distance charge transport in DNA:The hopping mechanism. Acc Chem Res, 2000,33:631-636
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7 Takada T, Kawai K, Fujitsuka M, Majima T. Direct observation of hole transfer through double-helical DNA over 100 A. PNAS,2004,101:14002-14006
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