📚 Pre-U CIE Biology: Quick-Reference Formula & Theorem Handbook | Pre-U CIE 生物：公式定理速查手册
This handbook compiles the essential formulas, equations, and quantitative principles that you must master for the Cambridge Pre-U Biology examination. Each section presents the key mathematical relationships along with concise explanations and worked contexts, helping you to apply them accurately in data analysis, experimental design, and theoretical questions.
本手册汇集了剑桥 Pre-U 生物学考试中必须掌握的核心公式、方程和定量原理。每个部分都给出了关键的数学关系，并附有简明解释和应用情境，帮助你在数据分析、实验设计和理论题中准确运用。
1. Magnification and Scale | 放大率与比例尺
Magnification links the size of an image to the actual size of the specimen.
放大率将图像尺寸与样本的实际尺寸联系起来。
Magnification = Image size / Actual size (M = I / A)
放大率 = 图像尺寸 / 实际尺寸 (M = I / A)
Ensure that both image size and actual size are expressed in the same unit. Common conversions: 1 mm = 1000 µm, 1 µm = 1000 nm.
务必保证图像尺寸和实际尺寸使用同一单位。常用换算：1 mm = 1000 µm，1 µm = 1000 nm。
When a scale bar is provided, measure its length on the image, determine the corresponding actual distance from the bar label, and then use the ratio to calculate actual dimensions of any structure.
当给出比例尺时，在图像上测量其长度，根据标注的对应实际距离，再按比例计算任意结构的实际尺寸。
To find actual size, rearrange the formula: A = I / M.
实际尺寸可由公式变形求得：A = I / M。
2. Surface Area to Volume Ratio | 表面积与体积比
The surface area to volume ratio (SA:V) governs the rate of exchange of substances across a boundary. Small cells or structures have a larger SA:V, favouring rapid diffusion.
表面积与体积比（SA:V）决定了物质跨越界面交换的速率。小细胞或结构具有较大的 SA:V，有利于快速扩散。
For a cube of side length l:
对于边长为 l 的立方体：
Surface area (SA) = 6l2 Volume (V) = l3
表面积 (SA) = 6l² 体积 (V) = l³
SA:V = 6 / l
As l increases, the SA:V decreases, limiting the efficiency of exchange. This principle explains why organisms need specialised respiratory and circulatory systems as body size increases.
随着 l 增大，SA:V 减小，交换效率受限。这一原理解释了为何随着体型增大，生物需要特化的呼吸和循环系统。
3. Mitotic Index | 有丝分裂指数
The mitotic index indicates the proportion of cells in a population that are undergoing mitosis. It is used to identify zones of active cell division in tissues such as root tips.
有丝分裂指数表示一个细胞群体中正在进行有丝分裂的细胞比例，可用于识别根尖等组织中细胞分裂活跃的区域。
Mitotic index = (Number of cells in mitosis / Total number of cells) × 100%
有丝分裂指数 = (有丝分裂细胞数 / 细胞总数) × 100%
Cells in any recognisable stage of mitosis (prophase, metaphase, anaphase, telophase) are counted. The total includes interphase cells as well.
处于任何可辨认的有丝分裂阶段（前期、中期、后期、末期）的细胞均计数在内，总数也包括间期细胞。
4. Hardy–Weinberg Equilibrium | 哈迪–温伯格平衡
The Hardy–Weinberg principle predicts allele and genotype frequencies in a large, randomly mating population that is not subject to evolutionary forces.
哈迪–温伯格定律可预测在一个无进化因素影响、随机交配的大种群中，等位基因和基因型频率的分布。
p + q = 1
p2 + 2pq + q2 = 1
Here, p is the frequency of the dominant allele, q is the frequency of the recessive allele; p2 is the frequency of homozygous dominant individuals, 2pq is the frequency of heterozygotes, and q2 is the frequency of homozygous recessives.
这里 p 为显性等位基因频率，q 为隐性等位基因频率；p² 为显性纯合子频率，2pq 为杂合子频率，q² 为隐性纯合子频率。
Use the equations to test whether a population is at equilibrium or to estimate carrier frequencies. Always start by calculating q from the observed recessive phenotype frequency.
利用这些方程可检验群体是否处于平衡状态，或估算携带者频率。通常从观测到的隐性表型频率入手计算 q。
5. Chi-squared (χ²) Test | 卡方（χ²）检验
The chi-squared test compares observed data with expected outcomes to determine whether any deviation is due to chance or a significant factor.
卡方检验通过比较观察数据与预期结果，判断偏差是偶然所致还是存在显著因素。
χ² = Σ (O − E)2 / E
O = observed value, E = expected value. The sum is taken over all categories.
O 为观察值，E 为预期值。求和涵盖所有分类。
Degrees of freedom (df) are calculated as (number of categories − 1) for simple goodness‑of‑fit tests. Compare the calculated χ² to the critical value at a chosen probability level (usually p = 0.05). If χ² > critical value, the null hypothesis is rejected.
对于简单的适合度检验，自由度 (df) 为分类数减 1。将计算所得的 χ² 值与选定概率水平（通常 p = 0.05）下的临界值比较，若 χ² > 临界值，则拒绝原假设。
6. Lincoln Index (Mark–Release–Recapture) | 林肯指数（标记-释放-重捕法）
The Lincoln index estimates population size of motile organisms using capture‑mark‑release‑recapture data. Several assumptions must hold for the estimate to be reliable.
林肯指数利用捕获-标记-释放-重捕的数据来估算活动型生物的种群大小。要使估算可靠，需满足若干假设。
N = (M × C) / R
N = estimated population size, M = number marked in the first sample, C = total caught in the second sample, R = number of marked individuals recaptured.
N 为估算种群大小，M 为首次标记个数，C 为第二次捕获总数，R 为重捕的标记个体数。
Assumptions include: no immigration or emigration, no births or deaths between sampling, marks are not lost and do not affect survival, and marked individuals mix randomly with the unmarked population.
假设包括：无迁入或迁出，两次取样之间无出生或死亡，标记不脱落且不影响存活，标记个体与未标记个体随机混合。
7. Respiratory Quotient (RQ) | 呼吸商（RQ）
Respiratory quotient is the ratio of carbon dioxide produced to oxygen consumed during respiration. The RQ value indicates the type of metabolic substrate being oxidised.
呼吸商是呼吸过程中产生的二氧化碳与消耗的氧气之比。RQ 值可揭示被氧化的代谢底物类型。
RQ = Volume of CO2 produced / Volume of O2 consumed
RQ = 产生的 CO₂ 体积 / 消耗的 O₂ 体积
Typical RQ values: carbohydrate ≈ 1.0, lipid ≈ 0.7, protein ≈ 0.9. During anaerobic respiration in plants, CO₂ may be released without oxygen uptake, preventing direct RQ calculation.
典型 RQ 值：碳水化合物约 1.0，脂质约 0.7，蛋白质约 0.9。植物无氧呼吸时可释放 CO₂ 而不摄取 O₂，此时无法通过该公式直接计算 RQ。
8. Productivity and Energy Transfer | 生产力与能量传递
Gross primary productivity (GPP) is the total energy fixed by autotrophs via photosynthesis. Net primary productivity (NPP) is the energy remaining after respiratory losses (R).
总初级生产力（GPP）是自养生物通过光合作用固定的总能量。净初级生产力（NPP）是减去呼吸损耗（R）后剩余的能量。
NPP = GPP − R
Energy transfer efficiency expresses the proportion of energy passed from one trophic level to the next.
能量传递效率表示能量从一个营养级传递到下一营养级的比例。
Efficiency (%) = (Energy in trophic level / Energy in previous trophic level) × 100
效率 (%) = (该营养级的能量 / 前一营养级的能量) × 100
GPP and NPP are often expressed in units such as kJ m⁻² yr⁻¹. Secondary productivity deals with energy incorporated by consumers.
GPP 和 NPP 常用单位如 kJ m⁻² yr⁻¹。次级生产力则涉及消费者同化的能量。
9. Water Potential (Ψ) in Plants | 植物水势（Ψ）
Water potential (Ψ) determines the direction of water movement. Water moves from regions of higher (less negative) water potential to regions of lower (more negative) water potential.
水势（Ψ）决定水分运动的方向。水从水势较高（负值较小）的区域流向水势较低（负值较大）的区域。
Ψ = Ψs + Ψp
Ψs is solute potential (always negative for dissolved solutes), and Ψp is pressure potential (positive inside turgid cells, zero at incipient plasmolysis, negative under tension in xylem).
Ψs 为溶质势（因溶质存在，恒为负值），Ψp 为压力势（在充盈细胞中为正值，质壁分离起始时为零，在木质部张力下为负值）。
At equilibrium, the water potential inside the cell equals that of the external solution. Gravitational potential is usually ignored in cellular‑level calculations.
平衡时，细胞内的水势与外界溶液的水势相等。在细胞水平计算中，通常忽略重力势。
10. Cardiac Output and Pulmonary Volumes | 心输出量与肺容量
Cardiac output represents the volume of blood pumped by each ventricle per minute, depending on heart rate and stroke volume.
心输出量表示每侧心室每分钟泵出的血液体积，取决于心率和每搏输出量。
Cardiac output (CO) = Heart rate (HR) × Stroke volume (SV)
心输出量 (CO) = 心率 (HR) × 每搏输出量 (SV)
Lung volumes and capacities follow specific relationships:
肺容积与肺容量有特定的关系：
| Abbreviation | Term | 中文 |
|---|---|---|
| TV | Tidal Volume | 潮气量 |
| IRV | Inspiratory Reserve Volume | 补吸气量 |
| ERV | Expiratory Reserve Volume | 补呼气量 |
| VC | Vital Capacity | 肺活量 |
| TLC | Total Lung Capacity | 肺总容量 |
VC = TV + IRV + ERV
TLC = VC + RV (Residual Volume)
Residual Volume (RV) is the air remaining after maximal exhalation and cannot be measured by spirometry alone.
残气量（RV）是最大呼气后仍残留于肺中的气体，无法单独通过肺活量计测量。
11. Standard Deviation and Standard Error | 标准差与标准误
Standard deviation measures the spread of data around the mean. Standard error indicates the precision of the mean estimate.
标准差衡量数据围绕均值的离散程度。标准误反映均值估计的精确度。
s = √[ Σ(x − x̄)2 / (n − 1) ]
s = sample standard deviation, x = individual value, x̄ = mean, n = sample size.
s 为样本标准差，x 为个体值，x̄ 为均值，n 为样本量。
SE = s / √n
The standard error (SE) is used to construct error bars on graphs; ±1 SE roughly represents a 68% confidence interval for the mean. Overlapping error bars suggest no significant difference, but formal statistical tests should be used to confirm.
标准误（SE）用于在图表上绘制误差棒；±1 SE 大约表示均值的 68% 置信区间。误差棒重叠提示差异不显著，但最终需依靠正式统计检验确认。
When comparing two means, a t‑test can then be performed using these values.
在比较两个均值时，可以利用这些数值进一步进行 t 检验。
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