Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Nucleic Acids

Field: Structural Biochemistry

Biopolymers, or large biomolecules, essential to all known forms of life.

Sequence of Expressions

Let B={A,T,G,C}B = \{A, T, G, C\} be the set of nucleobases. Define the pairing function P:B×B→{0,1}\mathcal{P}: B \times B \to \{0, 1\} such that P(b1,b2)=1\mathcal{P}(b_1, b_2) = 1 if b1b_1 and b2b_2 form a canonical pair, and P(b1,b2)=0\mathcal{P}(b_1, b_2) = 0 otherwise. The canonical pairing rules are defined by the constraints:\nP(A,T)=1,P(T,A)=1,P(G,C)=1,P(C,G)=1\mathcal{P}(A, T) = 1, \quad \mathcal{P}(T, A) = 1, \quad \mathcal{P}(G, C) = 1, \quad \mathcal{P}(C, G) = 1 \nAnd for all other pairs (b1,b2)∉{(A,T),(T,A),(G,C),(C,G)}(b_1, b_2) \notin \{(A, T), (T, A), (G, C), (C, G)\}, P(b1,b2)=0\mathcal{P}(b_1, b_2) = 0. This pairing dictates the formation of a stable base pair P=(b1,b2)P = (b_1, b_2).
Let SiS_i be the sugar unit and PiP_i be the phosphate group at position ii. The backbone is a polymer sequence defined by the repeating linkage Si−Pi−Si+1S_i - P_i - S_{i+1}. The phosphate group PiP_i links the 5′5'-hydroxyl group of SiS_i to the 3′3'-hydroxyl group of Si+1S_{i+1}. The chemical structure is formalized by the phosphodiester bond formation:\nSi−O−Pi−O−Si+1S_i - O - P_i - O - S_{i+1} \nwhere PiP_i is represented by the phosphate group PO2−\text{PO}_2^-. The backbone connectivity is defined by the sequence of linkages L={(S1,P1,S2),(S2,P2,S3),…,(SN−1,PN−1,SN)}\mathcal{L} = \{(S_1, P_1, S_2), (S_2, P_2, S_3), \dots, (S_{N-1}, P_{N-1}, S_N)\}. The overall charge density ρcharge\rho_{charge} is determined by the negative charges of the phosphate groups.
Let Σ={A,T,C,G}\Sigma = \{A, T, C, G\} be the finite alphabet of nucleotides. A nucleic acid sequence SS of length LL is defined as a vector S=(s1,s2,…,sL)S = (s_1, s_2, \dots, s_L), where si∈Σs_i \in \Sigma. The sequence can be represented mathematically as a word in the formal language Σ∗\Sigma^*. The sequence SS determines the chemical structure C(S)\mathcal{C}(S) and the associated thermodynamic properties ΔG(S)\Delta G(S) via the Hamiltonian H:ΣL→R\mathcal{H}: \Sigma^L \to \mathbb{R}.
Consider the polymerization reaction forming a polymer PP from NN monomers MM. Let [P][P] and [M][M] be the concentrations of the polymer chain and the monomer, respectively. The rate of polymerization RpolyR_{\text{poly}} is governed by the rate law: \nRpoly=kadd[P][M]/(1+Kinhib[I])R_{\text{poly}} = k_{\text{add}} [P] [M] / (1 + K_{\text{inhib}} [I]) \nwhere kaddk_{\text{add}} is the rate constant for monomer addition, and Kinhib[I]K_{\text{inhib}} [I] accounts for inhibition by side products [I][I]. The overall change in polymer length LL over time tt is given by the differential equation: \nd[P]dt=Rpoly\frac{d[P]}{dt} = R_{\text{poly}}
Let ri\mathbf{r}_i be the spatial coordinates of the ii-th base pair center, and let NN be the total number of base pairs. The double helix structure is defined by the coordinates R(i)\mathbf{R}(i) of the ii-th unit, parameterized by the helical index i∈Zi \in \mathbb{Z}. The coordinates must satisfy the following geometric constraints:\nR(i)=R0+ih(vz)+12π(vxcos⁡(2πiP)+vysin⁡(2πiP))\mathbf{R}(i) = \mathbf{R}_0 + \frac{i}{h} \left( \mathbf{v}_z \right) + \frac{1}{2\pi} \left( \mathbf{v}_x \cos\left(\frac{2\pi i}{P}\right) + \mathbf{v}_y \sin\left(\frac{2\pi i}{P}\right) \right) \nwhere PP is the helical pitch, hh is the rise per base pair, and vx,vy,vz\mathbf{v}_x, \mathbf{v}_y, \mathbf{v}_z are orthogonal unit vectors defining the axis and cross-section of the helix. Furthermore, the distance between paired bases must be constrained by the base pairing rules.
Consider two DNA strands, Strand1\text{Strand}_1 and Strand2\text{Strand}_2, represented by sequences of nucleotides indexed by ii. The directionality is defined by the indices ii. For Strand1\text{Strand}_1, the sequence runs from i=1i=1 to NN (the 5′→3′5' \to 3' direction). For Strand2\text{Strand}_2, the corresponding sequence must run from j=1j=1 to NN (the 5′→3′5' \to 3' direction). The antiparallel constraint requires that the ii-th nucleotide of Strand1\text{Strand}_1 pairs with the (N−i+1)(N-i+1)-th nucleotide of Strand2\text{Strand}_2. Mathematically, if B1(i)B_1(i) and B2(j)B_2(j) are the bases, then for a paired segment of length NN: \nB1(i) pairs with B2(N−i+1)B_1(i) \text{ pairs with } B_2(N-i+1) \nThis implies that the pairing index jj must be a linear function of the index ii such that j=N−i+1j = N-i+1.
Let SS be a sequence of length LL, and let P(si)P(s_i) be the probability of finding nucleotide sis_i at position ii. The information content I(S)I(S) stored in the sequence, measured in bits, is defined by the Shannon entropy HH: \nH(S)=−∑s∈ΣP(s)log⁡2P(s)H(S) = -\sum_{s \in \Sigma} P(s) \log_2 P(s) \nFor a sequence generated by a Markov process of order kk, the conditional probability P(si∣si−k,…,si−1)P(s_i | s_{i-k}, \dots, s_{i-1}) governs the sequence generation, and the total information is related to the joint probability distribution P(S)=∏i=1LP(si∣si−k,…,si−1)P(S) = \prod_{i=1}^{L} P(s_i | s_{i-k}, \dots, s_{i-1}).
Define the standard B-DNA helix by its helical parameters (αB,βB,γB)(\alpha_B, \beta_B, \gamma_B), where α\alpha is the rise per base pair, β\beta is the twist angle, and γ\gamma is the roll angle. The Z-DNA conformation is characterized by a transition to a left-handed helix and a zigzag backbone structure. Mathematically, this transition is defined by the change in the helical parameters: \nαZ=αB/2\alpha_Z = \alpha_B / 2 \nβZ=−βB\beta_Z = -\beta_B \nγZ=π/2\gamma_Z = \pi/2 \nThis transformation results in a backbone vector r(n)\mathbf{r}(n) that follows a zigzag path, deviating from the sinusoidal path of B-DNA.
Consider a donor-acceptor pair (D,A)(D, A) and a hydrogen bond interaction potential EHBE_{HB}. The potential energy EHBE_{HB} between a donor DD (e.g., N-H) and an acceptor AA (e.g., O or N) is modeled by a sum of distance and angular terms:\nEHB(r,θ)=E0(rDAr0)ne−k(rDA−r0)+Eθ(1−(cos⁡(θ−θ0))1)E_{HB}(r, \theta) = E_0 \left( \frac{r_{DA}}{r_0} \right)^n e^{-k(r_{DA} - r_0)} + E_{\theta} \left( 1 - \frac{(\cos(\theta - \theta_0))}{1} \right) \nwhere rDAr_{DA} is the distance between DD and AA, r0r_0 is the equilibrium distance, θ\theta is the angle formed by the donor-H-acceptor angle, and E0,Eθ,kE_0, E_{\theta}, k are empirical constants. Stable pairing requires EHB<EcritE_{HB} < E_{crit}.
Let S=(b1,b2,θ)S = (b_1, b_2, \boldsymbol{\theta}) be a stack of NN adjacent base pairs, where bib_i is the ii-th base pair and θ\boldsymbol{\theta} represents the relative orientation. The stacking interaction energy EstackE_{\text{stack}} is modeled as a pairwise potential sum over adjacent bases: \nEstack=12θTKθ+1N−1τTr(Ri,i+1)E_{\text{stack}} = \frac{1}{2} \boldsymbol{\theta}^T \boldsymbol{K} \boldsymbol{\theta} + \frac{1}{N-1} \boldsymbol{\tau} \text{Tr}\big(\boldsymbol{R}_{i, i+1}\big) \nwhere K\boldsymbol{K} is the stiffness matrix governing torsional strain, and Ri,i+1\boldsymbol{R}_{i, i+1} is the overlap matrix quantifying the π\pi-electron overlap between bib_i and bi+1b_{i+1}, typically approximated by a function of the distance di,i+1d_{i, i+1} and the dihedral angle ϕi,i+1\phi_{i, i+1}.