Edexcel Maths 6.3 Manipulating Genomes | Edexcel 数学 6.3 操作基因组

📚 Edexcel Maths 6.3 Manipulating Genomes | Edexcel 数学 6.3 操作基因组

Manipulating genomes is often seen as a topic in biology, but every stage of genome editing, sequencing and comparison relies on precise mathematical models. This unit explores the key mathematical tools used to represent DNA sequences, measure similarity, predict gene locations and assess editing risks.

操作基因组通常被视为生物学课题，但基因组编辑、测序和比较的每个阶段都依赖于精确的数学模型。本单元将探讨用于表示 DNA 序列、度量相似性、预测基因位置以及评估编辑风险的关键数学工具。

1. Genomes as Strings and Codes | 基因组作为字符串与编码

A DNA molecule can be represented as a string over the alphabet {A, C, G, T}. To apply mathematical operations, we often convert these letters into numerical codes such as A=0, C=1, G=2, T=3, which turns a genome into a sequence of integers.

DNA 分子可以表示为字母表 {A, C, G, T} 上的字符串。为了进行数学运算，我们通常将这些字母转换为数字编码，例如 A=0、C=1、G=2、T=3，从而把基因组变成整数序列。

Information theory then allows us to measure the complexity of a genome. For a sequence with base probabilities pₐ, p꜀, p𝓰, pₜ, the Shannon entropy is given by H = −(pₐ log₂pₐ + p꜀ log₂p꜀ + p𝓰 log₂p𝓰 + pₜ log₂pₜ), which quantifies the average information per base.

信息论允许我们度量基因组的复杂度。对于碱基概率为 pₐ、p꜀、p𝓰、pₜ 的序列，香农熵为 H = −(pₐ log₂pₐ + p꜀ log₂p꜀ + p𝓰 log₂p𝓰 + pₜ log₂pₜ)，它量化了每个碱基的平均信息量。

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2. Sequence Alignment and Edit Distance | 序列比对与编辑距离

To compare two DNA sequences, we need a measure of how different they are. The Levenshtein edit distance counts the minimum number of single-character insertions, deletions or substitutions needed to change one string into another.

为了比较两个 DNA 序列，我们需要一种度量它们差异的方法。莱文斯坦编辑距离计算将一个字符串变为另一个字符串所需的最少单字符插入、删除或替换次数。

If D(i, j) is the edit distance between the first i bases of sequence X and the first j bases of sequence Y, the recurrence is D(i, j) = min( D(i−1, j) + 1, D(i, j−1) + 1, D(i−1, j−1) + cost ), where cost is 0 if the bases match and 1 if they differ.

如果 D(i, j) 表示序列 X 的前 i 个碱基与序列 Y 的前 j 个碱基之间的编辑距离，递推公式为 D(i, j) = min( D(i−1, j) + 1, D(i, j−1) + 1, D(i−1, j−1) + cost )，其中当碱基匹配时 cost 为 0，不同时为 1。

This dynamic programming algorithm is the mathematical foundation of alignment tools such as BLAST and is widely used in genome manipulation to locate edited regions.

这种动态规划算法是 BLAST 等比对工具的数学基础，在基因组操作中广泛用于定位已编辑区域。

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3. Scoring Matrices and Similarity Measures | 打分矩阵与相似性度量

In real genomes, some substitutions occur more often than others. Scoring matrices such as PAM and BLOSUM assign a numerical score to each possible base or amino acid pair based on observed substitution frequencies.

在真实基因组中，某些替换比其他替换更常见。PAM 和 BLOSUM 等打分矩阵根据观察到的替换频率，为每对可能的碱基或氨基酸分配一个数值分数。

A typical log-odds score is S(a, b) = log₂( P(a, b) / (q(a) × q(b)) ), where P(a, b) is the probability that a and b are aligned in related sequences and q(a), q(b) are background frequencies.

典型的对数几率分数为 S(a, b) = log₂( P(a, b) / (q(a) × q(b)) )，其中 P(a, b) 是相关序列中 a 和 b 比对的概率，q(a)、q(b) 是背景频率。

Positive scores indicate that two bases appear together more often than chance, which helps identify homologous regions before editing.

正分数表示两个碱基共同出现的频率高于随机水平，这有助于在编辑前识别同源区域。

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4. Allele Frequencies and Hardy–Weinberg Equilibrium | 等位基因频率与哈迪-温伯格平衡

When manipulating a population of genomes, we track allele frequencies. For a gene with two alleles A and a, let p be the frequency of A and q = 1 − p be the frequency of a.

在操作一个群体的基因组时，我们需要跟踪等位基因频率。对于一个具有两个等位基因 A 和 a 的基因，设 p 为 A 的频率，q = 1 − p 为 a 的频率。

Under random mating and no selection, the genotype frequencies follow the Hardy–Weinberg equation p² + 2pq + q² = 1, where p², 2pq and q² give the frequencies of AA, Aa and aa.

在随机交配且没有选择的情况下，基因型频率遵循哈迪-温伯格方程 p² + 2pq + q² = 1，其中 p²、2pq 和 q² 分别给出 AA、Aa 和 aa 的频率。

To test whether observed genotype counts deviate from equilibrium, we use the chi-squared statistic χ² = Σ (O − E)² / E, where O is the observed count and E is the expected count for each genotype.

为了检验观察到的基因型计数是否偏离平衡，我们使用卡方统计量 χ² = Σ (O − E)² / E，其中 O 是每个基因型的观察计数，E 是期望计数。

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5. Sequencing Coverage and Poisson Distribution | 测序覆盖度与泊松分布

Before manipulating a genome, it must be sequenced. Coverage C is the average number of reads covering each base, calculated by the Lander–Waterman formula C = L × N / G, where L is read length, N is number of reads and G is genome size.

在操作基因组之前，必须先对其进行测序。覆盖度 C 是覆盖每个碱基的平均读数数目，可由兰德-沃特曼公式 C = L × N / G 计算，其中 L 是读长，N 是读数数量，G 是基因组大小。

The number of reads covering a particular base follows a Poisson distribution: P(X = n) = e⁻λ λⁿ / n!, where λ is the expected coverage. This allows us to estimate the probability of gaps in sequencing.

覆盖某一特定碱基的读数数量服从泊松分布：P(X = n) = e⁻λ λⁿ / n!，其中 λ 是期望覆盖度。这使我们能够估计测序中出现空白的概率。

High coverage reduces the chance that an edited site is missed, and the Poisson model helps choose N to achieve a target gap probability.

高覆盖度可降低遗漏编辑位点的机会，泊松模型有助于选择 N 以达到目标空白概率。

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6. Genome Rearrangements and Permutation Distance | 基因组重排与排列距离

Genome editing can cause large-scale rearrangements such as reversals and translocations. A chromosome segment can be modelled as a permutation of gene blocks, and the reversal distance is the minimum number of reversals needed to transform one permutation into another.

基因组编辑可能引起大规模重排，例如反转和易位。染色体片段可以建模为基因块的排列，反转距离是将一个排列转换为另一个排列所需的最少反转次数。

For a permutation, the number of inversions counts pairs of elements that are out of order. Sorting by reversals is a classic combinatorial problem with applications in comparing genomes across species.

对于一个排列，逆序数统计顺序颠倒的元素对。反转排序是一个经典的组合问题，在跨物种基因组比较中有广泛应用。

Algorithms such as the breakpoint graph method efficiently compute reversal distance, providing a metric for how many structural edits separate two genomes.

断点图等算法可以高效地计算反转距离，为两个基因组之间的结构编辑次数提供了一个度量。

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7. Hidden Markov Models for Gene Prediction | 基因预测中的隐马尔可夫模型

After editing, we need to identify coding regions. A hidden Markov model (HMM) treats the true biological state (exon, intron, intergenic) as a hidden sequence and the observed DNA bases as emissions.

编辑后，我们需要识别编码区域。隐马尔可夫模型（HMM）将真实生物学状态（外显子、内含子、基因间区）视为隐藏序列，将观察到的 DNA 碱基视为发射。

An HMM is defined by a transition matrix giving probabilities of moving between states and emission probabilities giving the chance of observing each base in a given state.

一个 HMM 由转移矩阵（给出状态之间转移的概率）和发射概率（给出在给定状态下观察每个碱基的机会）定义。

The Viterbi algorithm finds the most likely hidden state path using dynamic programming, and is a standard mathematical tool in genome annotation after manipulation.

维特比算法使用动态规划寻找最可能的隐藏状态路径，是基因组操作后进行注释的标准数学工具。

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8. Off-Target Effects and Probability in Gene Editing | 基因编辑脱靶效应的概率

CRISPR-Cas9 editing is guided by a short RNA sequence. It may bind to unintended sites, and the probability of such off-target effects can be modelled using sequence similarity and combinatorial counting.

CRISPR-Cas9 编辑由一个短 RNA 序列引导。它可能结合到非预期位点，这种脱靶效应的概率可以使用序列相似性和组合计数来建模。

If a guide sequence has length n, the number of possible off-target sites with up to m mismatches is given by the binomial sum Σᵣ₌₀ᵐ C(n, r) × 3ʳ, because each mismatch has three alternative bases.

如果引导序列长度为 n，最多有 m 个错配的可能脱靶位点数量由二项式和 Σᵣ₌₀ᵐ C(n, r) × 3ʳ 给出，因为每个错配有三个替代碱基。

Combining this with genome size and GC content gives an expected off-target count, helping researchers design safer guide RNAs.

将其与基因组大小和 GC 含量结合可得到预期的脱靶计数，帮助研究人员设计更安全的引导 RNA。

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9. Phylogenetic Trees and Distance Matrices | 系统发育树与距离矩阵

After editing several genomes, we can reconstruct their evolutionary relationships. A distance matrix stores pairwise genetic distances, and clustering algorithms such as UPGMA build a tree from these distances.

在编辑多个基因组后，我们可以重建它们的进化关系。距离矩阵存储成对的遗传距离，UPGMA 等聚类算法从这些距离构建系统发育树。

Species	A	B	C
A	0	0.12	0.18
B	0.12	0	0.20
C	0.18	0.20	0

UPGMA repeatedly joins the two closest clusters and recalculates distances as averages, producing a rooted tree with branch lengths proportional to divergence.

UPGMA 重复地将两个最接近的聚类合并，并将距离重新计算为平均值，产生一个分支长度与分歧程度成比例的有根树。

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10. Summary: How Mathematics Supports Genome Manipulation | 总结：数学如何支撑基因组操作

From string encodings and edit distances to Poisson coverage, HMMs, chi-squared tests and combinatorial off-target models, mathematics provides the quantitative core of genome manipulation.

从字符串编码和编辑距离，到泊松覆盖度、HMM、卡方检验以及组合脱靶模型，数学为基因组操作提供了定量核心。

Mastering these methods allows students to move beyond descriptive biology and perform rigorous analysis of sequence data, which is essential for modern genetic engineering.

掌握这些方法使学生能够超越描述性生物学，对序列数据进行严格分析，这对现代基因工程至关重要。

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