📚 Formula & Theorem Quick Reference Guide for Year 13 CAIE Biology | Year 13 CAIE 生物公式定理速查手册
This quick reference guide compiles the essential formulas, laws and quantitative concepts that every Year 13 CAIE Biology student must master. From Hardy–Weinberg equilibrium to ecological efficiency calculations, a solid command of these theorems will not only help you solve numerical problems but also deepen your understanding of core biological principles examined in Paper 4 and Paper 5.
这本速查手册汇集了每位 Year 13 CAIE 生物学生必须掌握的核心公式、定律与量化概念。从哈代–温伯格平衡到生态效率计算，熟练运用这些定理不仅能帮助你解决数据分析题，还能深化对 Paper 4 与 Paper 5 中核心生物原理的理解。
1. Hardy–Weinberg Principle | 哈代–温伯格定律
The Hardy–Weinberg principle states that in a large population where mating is random and there is no mutation, migration or natural selection, allele and genotype frequencies remain constant from generation to generation. This mathematical model shows what happens when NO evolutionary forces act on a population.
哈代–温伯格定律指出，在一个个体随机交配、没有突变、迁移和自然选择的大群体中，等位基因频率和基因型频率代代保持不变。该数学模型揭示了在没有进化力量作用时群体的遗传结构。
For a gene with two alleles, the frequencies are described by two equations:
对于一个具有两个等位基因的基因，频率可由两个方程描述：
p + q = 1
p² + 2pq + q² = 1
where p = frequency of the dominant allele, q = frequency of the recessive allele, p² = frequency of homozygous dominant genotype, 2pq = frequency of heterozygous genotype, and q² = frequency of homozygous recessive genotype. This is commonly used to calculate carrier frequencies in genetic disease contexts.
其中 p = 显性等位基因的频率，q = 隐性等位基因的频率，p² = 显性纯合子的频率，2pq = 杂合子的频率，q² = 隐性纯合子的频率。该公式常被用来计算遗传病的携带者频率。
2. Chi-Squared (χ²) Test | 卡方检验
The chi-squared test is used to determine whether there is a significant difference between observed and expected categorical data. In A2 Biology, it is most frequently applied to genetic crosses to test the goodness of fit between observed phenotypic ratios and those predicted by Mendelian genetics.
卡方检验用于判断观测数据与预期分类数据之间是否存在显著差异。在 A2 生物学中，该检验最常用于遗传杂交实验，以测验实际表型比例与孟德尔遗传预期比例的拟合优度。
χ² = Σ (O − E)² / E
O = observed value, E = expected value. The calculated χ² is compared against a critical value from a chi-squared distribution table, using the appropriate degrees of freedom (df = number of categories − 1). If χ² calculated > χ² critical, the null hypothesis is rejected, indicating a significant difference.
O = 观测值，E = 预期值。将计算得到的 χ² 值与卡方分布表中的临界值进行比较，自由度为 df = 类别数 − 1。若 χ² 计算值 > χ² 临界值，则拒绝原假设，表示存在显著差异。
3. Lincoln Index (Mark–Release–Recapture) | 林肯指数（标记重捕法）
The Lincoln index provides a method for estimating the population size of motile organisms. A sample is captured, marked and released; after allowing them to mix back into the population, a second sample is captured. The population size N is estimated using:
林肯指数用于估算活动范围较大的生物种群大小。首先捕获一样本进行标记后释放；待标记个体与种群充分混合后，再次捕获第二个样本。种群大小 N 的估算公式为：
N = (M × C) / R
where M = number of individuals marked in the first sample, C = total number captured in the second sample, and R = number of marked individuals recaptured in the second sample. Assumptions include no immigration, emigration, births or deaths between samples, and that marking does not affect survival or catchability.
其中 M = 首次标记个体数，C = 第二次捕获总数，R = 第二次捕获中带有标记的个体数。该方法假设两次取样期间没有迁入、迁出、出生与死亡，且标记不影响个体的存活或可捕性。
4. Simpson’s Diversity Index | 辛普森多样性指数
Simpson’s Index of Diversity (D) is a measure of biodiversity that accounts for both species richness and species evenness. A higher value of D indicates greater biodiversity.
辛普森多样性指数 (D) 是衡量生物多样性的指标，兼顾物种丰富度和均匀度。D 值越高，代表生物多样性越高。
D = 1 − Σ (n / N)²
n = total number of individuals of a particular species, N = total number of individuals of all species. In practice, calculate (n/N)² for each species, sum these values, then subtract from 1. This index is useful for comparing different habitats or assessing the impact of human activities on ecosystems.
n = 某一特定物种的个体总数，N = 所有物种的个体总数。实际计算时，先求每个物种的 (n/N)²，求和，再用 1 相减。该指数常用于比较不同栖息地或评估人类活动对生态系统的影响。
5. Respiratory Quotient (RQ) | 呼吸商
The respiratory quotient is the ratio of the volume of carbon dioxide produced to the volume of oxygen consumed during respiration. It provides information about the metabolic substrate being utilised.
呼吸商是细胞呼吸中产生的二氧化碳体积与消耗的氧气体积之比。它可以提示细胞正在利用哪类呼吸底物。
RQ = CO₂ produced / O₂ consumed
Typical RQ values: carbohydrate = 1.0, lipid ≈ 0.7, protein ≈ 0.9. A measured RQ above 1.0 often indicates anaerobic respiration, while values between 0.7 and 1.0 suggest a mixture of substrates. Respirometers are used to determine RQ experimentally.
典型的 RQ 值：糖类 = 1.0，脂质 ≈ 0.7，蛋白质 ≈ 0.9。实测 RQ 大于 1.0 常提示无氧呼吸，介于 0.7 到 1.0 之间则意味着混合底物。实验中可使用呼吸计测定 RQ。
6. Net Primary Production (NPP) & Ecological Efficiency | 净初级生产量与生态效率
In an ecosystem, the rate at which producers convert light energy into chemical energy is gross primary production (GPP). The energy remaining after plant respiration is net primary production, which represents the energy available to the next trophic level.
在生态系统中，生产者将光能转化为化学能的速率称为总初级生产量 (GPP)。扣去植物自身呼吸消耗后剩余的能量即为净初级生产量 (NPP)，它代表了可流向下一营养级的能量。
NPP = GPP − R
R = respiratory losses. The efficiency of energy transfer between trophic levels is usually calculated as:
R = 呼吸消耗。营养级之间的能量传递效率通常按下式计算：
Ecological efficiency (%) = (Energy transferred to next level / Energy available at previous level) × 100
These calculations highlight why food chains rarely exceed four or five trophic levels and underpin understanding of productivity in agricultural systems.
此类计算揭示了食物链通常不超过四到五个营养级的原因，也为理解农业系统的生产力提供了基础。
7. Water Potential (Ψ) | 水势
Water potential quantifies the tendency of water to move from one place to another. Water moves from regions of higher water potential to regions of lower water potential. It is a key concept for explaining water uptake by roots, movement through xylem and changes in guard cells.
水势用于量化水分由一处向另一处移动的倾向。水总是从高水势区域流向低水势区域。它是解释根部吸水、木质部输导以及保卫细胞变化的关键概念。
Ψ = Ψs + Ψp
Ψs = solute potential (always negative or zero; lowers water potential when solutes are present), Ψp = pressure potential (usually positive inside turgid cells, zero in an open solution). For a plant cell in pure water, Ψs may be, for example, −800 kPa and Ψp +400 kPa, giving a cell water potential of −400 kPa. The direction of water movement is always down a water potential gradient.
Ψs = 溶质势（总是为负值或零，溶质的存在会降低水势），Ψp = 压力势（在膨胀的细胞内通常为正值，在开放溶液中为零）。例如，一个植物细胞在纯水中，其 Ψs 可能为 −800 kPa，Ψp 为 +400 kPa，细胞水势即为 −400 kPa。水分运动的方向总是顺着水势梯度降低的方向。
8. Recombination Frequency & Gene Mapping | 重组频率与基因作图
Recombination frequency is the proportion of offspring that have a combination of alleles different from either parent, resulting from crossing over during meiosis. It is used to map the relative positions of genes on a chromosome.
重组频率是指在减数分裂过程中由于交叉互换而产生的、等位基因组合不同于任一亲本的后代所占的比例。它被用来绘制基因在染色体上的相对位置图谱。
Recombination frequency (%) = (Number of recombinant offspring / Total number of offspring) × 100
A recombination frequency of 1% corresponds to 1 map unit (centimorgan, cM). Linked genes show a recombination frequency less than 50%; unlinked genes (or genes far apart on the same chromosome) show approximately 50% recombination. A test cross with a double homozygous recessive is the standard way to measure recombination frequency.
1% 的重组频率相当于 1 个图距单位（厘摩，cM）。连锁基因的重组频率小于 50%；不连锁的基因（或同一条染色体上相距甚远的基因）重组频率约为 50%。测定重组频率的标准方法是与双隐性纯合子进行测交。
9. Magnification Calculations | 放大倍数计算
Magnification calculations are fundamental for interpreting microscope images and biological drawings. Even in Year 13, these skills are tested in planning and analysis questions, particularly where students must calibrate an eyepiece graticule or convert units.
放大倍数计算是解读显微图像和生物绘图的基础。即使在 Year 13，这些技能依然会在实验设计和数据分析题中考查，尤其涉及目镜测微尺的校准与单位换算。
Magnification = Image size / Actual size
Common unit conversions: 1 mm = 1000 µm, 1 µm = 1000 nm. When using a scale bar, determine the actual length it represents, then use the formula. Always express the answer to the same number of significant figures as the data and include units if required.
常用单位换算：1 mm = 1000 µm，1 µm = 1000 nm。若使用比例尺，首先确定其代表的实际长度，再代入公式。答案有效数字应与给定数据一致，并根据要求注明单位。
10. Rate Calculations for Biological Processes | 生物学过程的速率计算
Many exam questions require you to calculate the rate of a process such as enzyme activity, transpiration or population growth. The general rate formula is simple but must be tailored to the specific context.
许多考题要求计算某一过程的速率，例如酶活性、蒸腾作用或种群增长。通用的速率公式简单，但需根据具体情境灵活运用。
Rate = Change in quantity / Time taken
Examples: rate of enzyme reaction = (product formed) / time or (substrate used) / time, often expressed in units like μg min⁻¹ or mmol dm⁻³ s⁻¹. Rate of transpiration = distance moved by air bubble / time, used in potometer experiments. In population ecology, growth rate = (population change) / time period. Always pay attention to the units given in the question and any necessary conversions.
示例：酶反应速率 = (产物生成量) / 时间 或 (底物消耗量) / 时间，常用单位如 μg min⁻¹ 或 mmol dm⁻³ s⁻¹。蒸腾速率 = 气泡移动距离 / 时间，应用于蒸腾计实验。种群生态学中，增长率 = (种群变化量) / 时间段。务必留意题目所给的单位并进行必要的换算。
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