📚 Pre-U Cambridge Biology: Quick Reference Formula & Theorem Handbook | Pre-U 剑桥生物：公式定理速查手册
In the Cambridge Pre-U Biology course, quantitative skills are essential. This handbook compiles the key formulas, equations, and statistical tests that frequently appear in examinations, from population genetics to physiology. Mastery of these will strengthen data analysis and problem-solving abilities.
在剑桥 Pre-U 生物学课程中，定量技能至关重要。本手册汇集了从群体遗传学到生理学等考试中经常出现的关键公式、方程和统计检验。掌握这些内容将加强您的数据分析和问题解决能力。
1. Genetics and Population Genetics | 遗传学与群体遗传学
The Hardy–Weinberg principle predicts allele and genotype frequencies in a non-evolving population. For a gene with two alleles, the frequencies of the dominant (p) and recessive (q) alleles satisfy p + q = 1. The expected genotype frequencies are given by p² + 2pq + q² = 1, where p² is the frequency of the homozygous dominant genotype, 2pq of the heterozygous, and q² of the homozygous recessive.
哈代-温伯格定律预测非进化群体中的等位基因和基因型频率。对于具有两个等位基因的基因，显性 (p) 和隐性 (q) 等位基因的频率满足 p + q = 1。预期的基因型频率由 p² + 2pq + q² = 1 给出，其中 p² 是纯合显性基因型的频率，2pq 是杂合子频率，q² 是纯合隐性频率。
The chi-squared (χ²) test is used to compare observed and expected frequencies. The formula is:
χ² = Σ((O – E)² / E)
where O is the observed value and E is the expected value. A significant difference suggests that the population is not in Hardy–Weinberg equilibrium.
卡方 (χ²) 检验用于比较观测频率和预期频率。公式为 χ² = Σ((O – E)² / E)，其中 O 是观测值，E 是预期值。显著性差异表明群体未处于哈代-温伯格平衡。
Recombination frequency measures the proportion of recombinant offspring. It is calculated as:
Recombination frequency (%) = (Number of recombinant progeny / Total number of progeny) × 100%
One map unit (centimorgan) corresponds to a 1% recombination frequency.
重组频率衡量重组子代的比例。计算公式为 重组频率 (%) = (重组子代数 / 总子代数) × 100%。一个图距单位 (厘摩) 相当于 1% 的重组频率。
2. Population Ecology | 种群生态学
Exponential growth occurs when resources are unlimited. The rate of population change is dN/dt = rN, where N is population size and r is the intrinsic rate of increase. The population size at time t is given by:
Nt = N0 ert
指数增长发生在资源无限的条件下。种群变化速率为 dN/dt = rN，其中 N 为种群大小，r 为内禀增长率。在时间 t 时的种群大小由 Nt = N0 ert 给出。
The logistic growth model incorporates carrying capacity (K). The growth rate is given by:
dN/dt = rN(1 – N/K)
As N approaches K, the growth rate slows and eventually stops.
逻辑斯谛增长模型纳入了环境容纳量 (K)。增长速率为 dN/dt = rN(1 – N/K)。当 N 接近 K 时，增长速率减慢并最终停止。
The Lincoln index estimates population size using mark-release-recapture data. The formula is:
N = (M × C) / R
where M is the number initially marked, C is the total caught in the second sample, and R is the number of marked individuals recaptured.
林肯指数利用标记-释放-重捕数据估计种群大小。公式为 N = (M × C) / R，其中 M 是初始标记数，C 是第二次捕获的总数，R 是重捕的标记个体数。
3. Microscopy and Magnification | 显微镜与放大率
Magnification is the ratio of image size to actual size. The formula is:
Magnification = Image size / Actual size
Ensure both measurements are in the same units. A scale bar can be used to determine the actual size of a structure.
放大率是图像大小与实际大小之比。公式为 放大率 = 图像大小 / 实际大小。确保两个测量值使用相同的单位。比例尺可用于确定结构的实际大小。
To calculate actual size from a scale bar, use: Actual size = Measured length of scale bar / Magnification of scale bar. Often, the scale bar represents a known length such as 10 µm.
根据比例尺计算实际大小时，使用：实际大小 = 比例尺测量长度 / 比例尺放大率。通常比例尺代表已知长度，如 10 µm。
4. Surface Area to Volume Ratio | 表面积与体积比
The surface area to volume ratio (SA:V) determines the efficiency of exchange with the environment. It is calculated as:
SA:V = Surface area / Volume
For a cube of side length L, SA = 6L² and V = L³, giving SA:V = 6/L. As organism size increases, the SA:V decreases, restricting diffusion and necessitating specialised transport systems.
表面积与体积比 (SA:V) 决定了与环境交换的效率。计算公式为 SA:V = 表面积 / 体积。对于边长为 L 的立方体，SA = 6L²，V = L³，因此 SA:V = 6/L。随着生物体增大，SA:V 减小，限制了扩散，从而需要专门的运输系统。
5. Transport and Exchange Formulae | 运输与交换公式
Fick’s Law quantifies the rate of diffusion across a membrane:
Rate of diffusion = (Surface area × Concentration gradient) / Thickness of membrane
Increasing surface area or concentration gradient, or decreasing membrane thickness, enhances diffusion rate.
菲克定律定量描述跨膜扩散速率：扩散速率 = (表面积 × 浓度梯度) / 膜厚度。增加表面积或浓度梯度，或降低膜厚度，均可提高扩散速率。
Cardiac output (CO) is the volume of blood pumped by the heart per minute:
CO = Heart rate × Stroke volume
单位：CO 通常以 L/min 表示。心脏通过调节心率或每搏输出量来改变输出量。
Pulmonary ventilation rate (minute volume) is calculated as:
Minute volume = Tidal volume × Breathing rate
Tidal volume is the volume of air moved in and out of the lungs during a normal breath. This relationship is fundamental in respiratory physiology.
肺通气量（分钟通气量）计算为：分钟通气量 = 潮气量 × 呼吸频率。潮气量是正常呼吸时空气进出肺部的体积。这一关系是呼吸生理学的基础。
6. Water Potential | 水势
Water potential (ψ) dictates the direction of water movement. The formula in plant cells is:
ψ = ψs + ψp
where ψs is the solute potential (always negative or zero) and ψp is the pressure potential (usually positive). Water flows from regions of higher water potential to lower water potential.
水势 (ψ) 决定了水分运动的方向。植物细胞中的公式为 ψ = ψs + ψp，其中 ψs 是溶质势（始终为负或零），ψp 是压力势（通常为正）。水从水势较高的区域流向水势较低的区域。
At incipient plasmolysis, pressure potential is zero, so ψ = ψs. Fully turgid cells have ψ = 0. This equation helps explain turgor maintenance and wilting.
在初始质壁分离时，压力势为零，因此 ψ = ψs。完全膨胀的细胞 ψ = 0。该方程有助于解释膨压的维持和萎蔫现象。
7. Statistical Analysis | 统计分析
The chi-squared (χ²) test is widely used in biology to compare categorical data. The formula is:
χ² = Σ((O – E)² / E)
Degrees of freedom (df) = number of categories – 1. The calculated χ² is compared to a critical value at a chosen probability level (usually p = 0.05) to determine significance.
卡方 (χ²) 检验在生物学中广泛用于比较分类数据。公式为 χ² = Σ((O – E)² / E)。自由度 (df) = 类别数 – 1。将计算所得的 χ² 值与选定概率水平（通常 p = 0.05）的临界值进行比较，以确定显著性。
Standard error of the mean (SE) indicates the precision of a sample mean:
SE = s / √n
where s is the sample standard deviation and n is the sample size. The 95% confidence interval for the population mean is approximately:
95% CI = x̄ ± 1.96 × SE
These are used when comparing means in biological investigations.
标准误差 (SE) 表示样本均值的精确度：SE = s / √n，其中 s 为样本标准差，n 为样本大小。总体均值的 95% 置信区间近似为：95% CI = x̄ ± 1.96 × SE。在生物学研究中比较均值时使用。
8. Energy and Respiratory Quotient | 能量与呼吸商
Respiratory quotient (RQ) is the ratio of carbon dioxide produced to oxygen consumed during respiration:
RQ = Volume of CO₂ produced / Volume of O₂ consumed
RQ values indicate the respiratory substrate: carbohydrate ~1.0, lipid ~0.7, protein ~0.9.
呼吸商 (RQ) 是呼吸过程中产生二氧化碳与消耗氧气的体积比：RQ = 产生的 CO₂ 体积 / 消耗的 O₂ 体积。RQ 值指示呼吸底物：碳水化合物的 RQ 约为 1.0，脂质约为 0.7，蛋白质约为 0.9。
Energy content of biological material can be estimated using a calorimeter. The formula for energy released is:
Energy (kJ) = Mass of sample (g) × Energy content per unit mass (kJ g⁻¹)
This is used in ecological studies and food testing.
生物材料的能量含量可用量热计估算。释放的能量公式为：能量 (kJ) = 样品质量 (g) × 每单位质量能量含量 (kJ g⁻¹)。这用于生态学研究和食品检测。
9. Growth Kinetics and Doubling Time | 生长动力学与倍增时间
Bacterial growth in batch culture follows exponential growth when nutrients are plentiful. The number of cells after n divisions is:
Nt = N0 × 2n
where
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