📚 Year 12 Cambridge Biology: Formula & Theorem Quick-Reference Handbook | Year 12 Cambridge 生物：公式定理速查手册
This quick-reference handbook gathers all essential formulas, equations, laws, theorems, and key principles you need for Cambridge AS Level Biology (9700). It is designed to help you revise quantitative skills and core biological concepts efficiently. Each entry is presented in a paired bilingual format to strengthen your understanding and exam readiness.
本速查手册汇集了剑桥 AS 生物（9700）所有必备的公式、方程、定律、定理和核心原理，旨在帮助您高效复习量化技能与核心生物学概念。每个条目均以配对中英双语呈现，以加深理解并提升应试能力。
1. Microscopy & Magnification Calculations | 显微镜与放大率计算
The magnification equation is fundamental for calibrating microscope images: Magnification = Image size ÷ Actual size. Always ensure that both quantities are expressed in the same unit. The relationship can be rearranged as Actual size = Image size ÷ Magnification or Image size = Magnification × Actual size.
放大倍率公式是校准显微镜图像的基础：放大倍率 = 图像大小 ÷ 实际大小。务必确保两个量为同一单位。该关系可变形为：实际大小 = 图像大小 ÷ 放大倍率 或 图像大小 = 放大倍率 × 实际大小。
Common unit conversions are essential: 1 mm = 1000 µm and 1 µm = 1000 nm. When working with scale bars, first measure the bar length in mm or µm, then apply the magnification formula to find either the actual size of a structure or the drawing magnification.
常用单位换算是关键：1 mm = 1000 µm，1 µm = 1000 nm。处理比例尺时，先测量尺条长度（mm 或 µm），再套用放大率公式求出结构实际大小或绘图放大倍率。
Magnification = I / A
2. Fick’s Law of Diffusion | 菲克扩散定律
Fick’s law describes the rate of diffusion across an exchange surface: Rate of diffusion ∝ (Surface area × Concentration difference) ÷ Thickness of exchange surface (diffusion distance). To maximise the rate, organisms have evolved large, thin, and highly vascularised surfaces that maintain a steep concentration gradient.
菲克定律描述了物质在交换表面上的扩散速率：扩散速率 ∝（表面积 × 浓度差）÷ 交换表面厚度（扩散距离）。为了使速率最大化，生物体演化出了大面积、极薄且高度血管化的表面，以维持陡峭的浓度梯度。
Adaptations such as alveoli in lungs, gill filaments in fish, and spongy mesophyll in leaves all align with Fick’s law. For example, the alveolar wall is one cell thick, providing a minimal diffusion distance, while constant ventilation and blood flow renew the concentration gradient of oxygen and carbon dioxide.
肺泡、鱼鳃丝和叶的海绵组织等适应性结构均符合菲克定律。例如，肺泡壁仅一个细胞之厚，提供了极短的扩散距离，而持续的通风和血流更新了氧气与二氧化碳的浓度梯度。
3. Hardy-Weinberg Principle | 哈代-温伯格定律
The Hardy-Weinberg principle provides a mathematical model to study allele and genotype frequencies in a non-evolving population. It is expressed as two equations: p + q = 1 and p² + 2pq + q² = 1, where p represents the frequency of the dominant allele, and q represents the frequency of the recessive allele. The homozygous dominant genotype frequency is p², heterozygous is 2pq, and homozygous recessive is q².
哈代-温伯格定律提供了一个数学工具，用于研究不进化群体中的等位基因频率和基因型频率。其表达式为 p + q = 1 和 p² + 2pq + q² = 1，其中 p 为显性等位基因频率，q 为隐性等位基因频率。纯合显性基因型频率为 p²，杂合子为 2pq，纯合隐性为 q²。
For the principle to hold, five conditions must be met: large population size, random mating, no mutations, no gene flow (migration), and no natural selection. Any deviation from these conditions indicates that microevolution is occurring. In exam questions, you are often given the frequency of the homozygous recessive phenotype (q²) and asked to calculate allele and carrier frequencies.
该定律成立需要满足五个条件：大群体、随机交配、无突变、无基因流动（迁移）和无自然选择。任何偏离都意味着微进化正在发生。考试中常给出纯合隐性表型的频率（q²），要求计算等位基因频率和携带者频率。
4. Chi-Squared (χ²) Test | 卡方检验
The chi-squared test evaluates whether the difference between observed and expected results is due to chance. The formula is: χ² = Σ (O − E)² ÷ E, where O is the observed value and E is the expected value for each category. The larger the χ² value, the greater the deviation from the null hypothesis.
卡方检验用于评估观察值与预期值之间的差异是否由偶然造成。公式为：χ² = Σ (O − E)² ÷ E，其中 O 为每类的观察值，E 为预期值。χ² 值越大，与零假设的偏离就越大。
Degrees of freedom (df) = number of categories − 1. Compare the calculated χ² against the critical value at a probability level of p = 0.05. If χ² calculated > critical value, reject the null hypothesis, meaning the difference is statistically significant and not due to chance alone. This test is frequently applied to genetic ratios (e.g., 3:1, 9:3:3:1).
自由度（df）= 类别数 − 1。将计算出的 χ² 值与 p = 0.05 概率水平下的临界值比较。若 χ² 计算值 > 临界值，则拒绝零假设，表明差异具有统计显著性，并非仅由偶然引起。该检验常用于遗传比例（如 3:1、9:3:3:1）。
χ² = Σ (O − E)² / E
5. Mendelian Inheritance & Probability Rules | 孟德尔遗传与概率法则
Mendel’s first law, the Law of Segregation, states that each individual has two alleles for a gene, and these alleles segregate during gamete formation so that each gamete carries only one allele. The second law, the Law of Independent Assortment, asserts that alleles of different genes assort independently during meiosis, provided the genes are on different chromosomes.
孟德尔第一定律——分离定律——指出个体每个基因有两个等位基因，在配子形成过程中等位基因分离，每个配子只含一个等位基因。第二定律——自由组合定律——强调不同基因的等位基因在减数分裂中独立分配，前提是这些基因位于不同染色体上。
Probability rules facilitate genetic predictions: the multiplication rule states that the probability of two independent events occurring together is the product of their individual probabilities, while the addition rule says that the probability of any one of several mutually exclusive events equals the sum of their individual probabilities. These apply to monohybrid and dihybrid crosses, yielding classic phenotypic ratios of 3:1 and 9:3:3:1.
概率法则辅助遗传预测：乘法法则指出两个独立事件同时发生的概率等于各自概率的乘积，加法法则指出若干互斥事件中任一事件发生的概率等于各自概率之和。这适用于单因子和双因子杂交，产生经典表型比 3:1 和 9:3:3:1。
6. Water Potential (ψ) | 水势（ψ）
Water potential (ψ) determines the direction of water movement by osmosis. It is the sum of solute potential (ψₛ) and pressure potential (ψₚ): ψ = ψₛ + ψₚ. Water moves from regions of higher (less negative) water potential to regions of lower (more negative) water potential.
水势（ψ）决定了渗透作用中水的运动方向。它是溶质势（ψₛ）与压力势（ψₚ）的总和：ψ = ψₛ + ψₚ。水从水势较高（较不负）的区域流向水势较低（较负）的区域。
Solute potential (ψₛ) is always negative or zero because dissolved solutes reduce the free energy of water; the more solute, the more negative the ψₛ. Pressure potential (ψₚ) is typically positive inside plant cells due to turgor pressure pushing against the cell wall. When ψₚ equals the absolute value of ψₛ, the cell is at equilibrium and ψ = 0; this explains turgid, flaccid, and plasmolyzed states in plant cells.
溶质势（ψₛ）总是负值或零，因为溶解的溶质降低了水的自由能；溶质越多，ψₛ 越负。压力势（ψₚ）在植物细胞内通常为正值，因为膨压推挤细胞壁。当 ψₚ 等于 ψₛ 的绝对值时，细胞处于平衡且 ψ = 0；这解释了植物细胞中的挺立、萎蔫和质壁分离状态。
ψ = ψₛ + ψₚ
7. Cell Theory | 细胞学说
The cell theory is a fundamental unifying concept in biology stating that all living organisms are composed of one or more cells, the cell is the basic structural and functional unit of life, and all cells arise from pre-existing cells by division. This theory was refined through contributions from Schleiden, Schwann, and Virchow.
细胞学说是生物学中一个根本的统一概念，提出：所有生物体均由一个或多个细胞构成，细胞是生命的基本结构和功能单位，所有细胞均来自已存在的细胞分裂。施莱登、施旺和菲尔绍的贡献完善了这一学说。
Exceptions to the cell theory include viruses, which lack cellular structure and cannot replicate independently, and the origin of the first cell, which must have arisen from non-living material through abiogenesis. Nonetheless, the theory underpins modern biology, linking microscopy, genetics, and biochemistry.
不符合细胞学说的例外包括病毒（缺乏细胞结构且不能独立复制）以及第一个细胞的起源（必定通过非生命物质的自然发生而来）。尽管如此，该学说仍是现代生物学的基础，连接了显微镜学、遗传学与生物化学。
8. Fluid Mosaic Model of Cell Membrane | 细胞膜流动镶嵌模型
The fluid mosaic model describes the structure of cell membranes as a phospholipid bilayer with embedded proteins, cholesterol, and glycoproteins. The ‘fluid’ aspect refers to the lateral movement of phospholipids and proteins, while ‘mosaic’ refers to the patchwork of different proteins floating in the bilayer.
流动镶嵌模型将细胞膜描述为磷脂双分子层，其中嵌有蛋白质、胆固醇和糖蛋白。”流动”指磷脂和蛋白质的侧向运动，”镶嵌”指双层中各种蛋白质拼缀在一起的图案。
Phospholipids are amphipathic, with hydrophilic phosphate heads facing the aqueous interior and exterior and hydrophobic fatty acid tails facing inward. Integral proteins act as channels or carriers, peripheral proteins are involved in signalling, and cholesterol modulates membrane fluidity. This model explains selective permeability, cell recognition, and signal transduction.
磷脂是双亲性分子，亲水磷酸头朝向水相内外部，疏水脂肪酸尾朝内。整合蛋白充当通道或载体，外周蛋白参与信号传导，胆固醇调节膜流动性。该模型解释了选择性通透、细胞识别和信号转导。
9. Theory of Natural Selection | 自然选择学说
The theory of natural selection, proposed by Darwin and Wallace, explains how populations evolve. It states that there is genetic variation within a population, organisms produce more offspring than can survive, there is competition for limited resources, and individuals with heritable traits better suited to the environment are more likely to survive and reproduce. Over generations, advantageous alleles increase in frequency, leading to adaptation.
达尔文与华莱士提出的自然选择学说解释了种群如何进化。该学说指出：种群内存在遗传变异，生物产生的后代多过能存活的数量，有限资源引发竞争，那些具有更适合环境遗传性状的个体更易存活并繁殖。经过数代，有利等位基因频率增加，导致适应。
Modern examples include the evolution of antibiotic resistance in bacteria and changes in allele frequency of the peppered moth during the Industrial Revolution. Natural selection acts on phenotypes, but it is the genotype that is inherited. The theory requires the principles of inheritance and variation to operate effectively.
现代例子包括细菌抗生素抗性的进化以及工业革命期间桦尺蛾等位基因频率的改变。自然选择作用于表型，但遗传的是基因型。该学说需要遗传和变异原理才能有效运作。
10. Endosymbiotic Theory | 内共生学说
The endosymbiotic theory proposes that mitochondria and chloroplasts originated from free-living prokaryotes that were engulfed by ancestral eukaryotic cells. Instead of being digested, the engulfed bacteria formed a symbiotic relationship, eventually evolving into organelles. Evidence includes the presence of double membranes, their own circular DNA, 70S ribosomes, and the ability to replicate independently by binary fission.
内共生学说指出，线粒体和叶绿体起源于被原始真核细胞吞噬的自由生活原核生物。被吞噬的细菌未被消化，而是形成共生关系，最终演化成为细胞器。支持证据包括双膜结构、自身的环状 DNA、70S 核糖体以及通过二分裂独立复制的能力。
This theory explains why these two organelles share more biochemical characteristics with bacteria than with eukaryotic cells, such as sensitivity to antibiotics like chloramphenicol. It is a cornerstone of our understanding of eukaryotic cell evolution and the complexity of life.
该学说解释了为何这两种细胞器在生化特征上更接近细菌而非真核细胞，例如对氯霉素等抗生素敏感。它是我们理解真核细胞进化和生命复杂性的基石。
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