📚 Pre-U AQA Biology: Formula & Theorem Quick Reference | Pre-U AQA 生物：公式定理速查手册
This quick reference guide compiles all essential formulas and theorems required for the AQA Pre-U Biology examination. Mastery of these quantitative tools is vital for tackling data-analysis questions, genetics problems, and ecology calculations. Accompanying explanations help you understand the context and assumptions behind each equation.
本速查手册汇编了AQA Pre-U生物学考试所需的所有关键公式和定理。掌握这些定量工具对于处理数据分析题、遗传学问题和生态学计算至关重要。附带的解释有助于理解每个方程的背景和假设。
1. Hardy-Weinberg Principle | 哈代-温伯格定律
The Hardy-Weinberg principle acts as a null hypothesis for population genetics. It predicts that allele and genotype frequencies remain constant across generations if no evolutionary forces act. The two fundamental equations are the allele frequency equation and the genotype frequency equation.
哈代-温伯格原理充当群体遗传学的零假设。它预测如果没有进化力量的作用，等位基因频率和基因型频率将在世代间保持恒定。两个基本方程是等位基因频率方程和基因型频率方程。
p + q = 1
p² + 2pq + q² = 1
Here p represents the frequency of the dominant allele, while q stands for the frequency of the recessive allele. The term p² gives the frequency of homozygous dominant individuals, 2pq the heterozygotes, and q² the homozygous recessive individuals.
这里 p 代表显性等位基因的频率，而 q 代表隐性等位基因的频率。p² 项给出显性纯合子的频率，2pq 为杂合子频率，q² 为隐性纯合子的频率。
If the observed genotype numbers in a population deviate significantly from the expected frequencies calculated from allele frequencies, then one or more assumptions (no mutation, no migration, large population, random mating, no selection) are likely violated.
如果一个种群中观察到的基因型数量显著偏离根据等位基因频率计算出的预期频率，那么一个或多个假设（无突变、无迁移、大种群、随机交配、无选择）很可能被违反。
2. Chi-Squared Test | 卡方检验
The chi-squared (χ²) test is used in biology to compare observed results with expected results and determine whether any difference is due to chance or a statistically significant factor. It is commonly applied to genetic crosses and ecological distribution studies.
卡方（χ²）检验在生物学中用于比较观测结果与预期结果，并判断任何差异是由偶然引起还是由统计显著因素引起。它常应用于遗传杂交和生态分布研究。
χ² = Σ((O − E)² ÷ E)
O represents the observed frequency and E the expected frequency for each category. The summation runs over all categories. The larger the calculated χ² value, the greater the discrepancy between observed and expected data.
O 代表每个类别的观测频数，E 为预期频数。对所有类别求和。计算出的 χ² 值越大，观测数据与预期数据之间的差异越大。
Compare the calculated χ² against a critical value from the χ² distribution table at a chosen probability level (usually p = 0.05) and the appropriate degrees of freedom (d.f. = number of categories − 1). If χ² > critical value, reject the null hypothesis.
将计算出的 χ² 与选定概率水平（通常 p = 0.05）和适当自由度（d.f. = 类别数 − 1）下 χ² 分布表中的临界值进行比较。若 χ² > 临界值，则拒绝零假设。
3. Simpson’s Diversity Index | 辛普森多样性指数
Simpson’s Diversity Index (D) quantifies the biodiversity of a habitat. A higher value indicates greater diversity, meaning the community is dominated by many different species rather than a single one. The index accounts for both species richness and evenness.
辛普森多样性指数（D）量化栖息地的生物多样性。数值越高表示多样性越大，意味着群落由许多不同物种主导，而非单一物种。该指数兼顾物种丰富度和均匀度。
D = 1 − Σ(n ÷ N)²
In this formula, n is the total number of individuals of a particular species, and N is the total number of individuals of all species. The sum is taken over all species present. D can range from 0 (no diversity) to almost 1 (infinite diversity).
在此公式中，n 是某一特定物种的个体总数，N 是所有物种的个体总数。对所有存在的物种求和。D 的范围可以从 0（无多样性）到接近 1（无限多样性）。
Alternatively, the reciprocal form 1/Σ(n/N)² is sometimes used; be consistent with the exam specification. AQA Pre-U expects candidates to calculate D and interpret the result, such as discussing the impact of habitat degradation on biodiversity.
有时也使用倒数形式 1/Σ(n/N)²；应遵循考试大纲。AQA Pre-U 要求考生计算 D 并解释结果，例如讨论栖息地退化对生物多样性的影响。
4. Lincoln Index (Capture-Mark-Recapture) | 林肯指数（标记重捕法）
The Lincoln index estimates the population size of mobile organisms. A sample is captured, marked, and released. Later a second sample is taken, and the proportion of marked individuals recaptured provides the estimate.
林肯指数估算移动生物的种群大小。先捕获、标记并释放一个样本。随后采集第二个样本，重捕的标记个体比例即可提供估计值。
N = (M × C) ÷ R
Here N = estimated total population, M = number of individuals caught and marked in the first sample, C = total number caught in the second sample, and R = number of marked individuals recaptured in the second sample.
这里 N = 估算的总种群数量，M = 第一次样本中捕获并标记的个体数，C = 第二次样本中捕获的总数，R = 第二次样本中重捕的标记个体数。
The method relies on several assumptions: marks do not harm or disadvantage the animals, marks are not lost, marked individuals mix randomly, the population is closed (no births, deaths, immigration, or emigration), and capture probability is equal for all.
该方法依赖于几个假设：标记不伤害动物或使其处于劣势，标记不脱落，标记个体随机混合，种群是封闭的（无出生、死亡、迁入或迁出），且所有个体的捕获概率相等。
5. Population Growth Models | 种群增长模型
Exponential growth occurs when resources are unlimited. The rate of population increase is proportional to the current population size. This scenario is described by the simple differential equation.
当资源无限时发生指数增长。种群增长率与当前种群大小成正比。此情景由简单的微分方程描述。
dN/dt = rN
N is the population size, t is time, and r is the intrinsic rate of natural increase (birth rate minus death rate). The solution predicts a J-shaped curve. Real populations cannot sustain exponential growth indefinitely.
N 是种群大小，t 是时间，r 是内禀自然增长率（出生率减去死亡率）。其解预测出 J 形曲线。现实种群无法无限维持指数增长。
Logistic growth incorporates environmental resistance through a carrying capacity (K). The growth rate slows as the population approaches K, yielding an S-shaped sigmoid curve.
逻辑斯蒂增长通过环境容纳量（K）纳入环境阻力。随着种群接近 K，增长率放缓，产生 S 形曲线。
dN/dt = rN((K − N) ÷ K)
When N is small, (K−N)/K ≈ 1, so growth approximates exponential. When N = K, the term becomes zero and population size stabilises. This model is more realistic for populations in limited habitats.
当 N 很小时，(K−N)/K ≈ 1，因此增长近似于指数。当 N = K 时，该项变为零，种群大小趋于稳定。该模型对于有限栖息地中的种群更现实。
6. Cardiac Output | 心输出量
Cardiac output (CO) is the volume of blood pumped by one ventricle of the heart per minute. It determines the rate at which oxygen is delivered to tissues and is a key measure of cardiovascular fitness.
心输出量（CO）是心脏一个心室每分钟泵出的血液体积。它决定了向组织输送氧气的速率，是衡量心血管健康的关键指标。
CO = HR × SV
HR is heart rate in beats per minute (bpm), and SV is stroke volume, the volume of blood ejected per beat (typically in mL or L). Units of CO are commonly L/min. At rest, a typical cardiac output is around 5 L/min.
HR 是心率，单位为每分钟心跳数（bpm），SV 是每搏输出量，即每次心跳射出的血液体积（通常以 mL 或 L 表示）。CO 的单位通常为 L/min。静息时，典型的心输出量约为 5 L/min。
During exercise, both heart rate and stroke volume increase, raising cardiac output to deliver more oxygen to muscles. Training can increase stroke volume, leading to a lower resting heart rate for the same cardiac output.
运动时，心率和每搏输出量均增加，提高心输出量以为肌肉输送更多氧气。训练可增加每搏输出量，从而在相同心输出量下降低静息心率。
7. Fick’s Law of Diffusion | 菲克扩散定律
Fick’s Law summarises the factors that affect the rate of diffusion across a membrane or tissue. It is especially relevant for gas exchange in lungs, gills, and leaves, as well as nutrient absorption in the small intestine.
菲克定律总结了影响跨膜或组织扩散速率的因素。它与肺、鳃和叶片中的气体交换以及小肠中的营养吸收尤其相关。
Rate of diffusion ∝ (Surface area × Concentration difference) ÷ Thickness of exchange surface
A larger surface area and a steeper concentration gradient increase the diffusion rate, while a greater diffusion distance (thickness) decreases it. The law can be written as R ∝ (A × ΔC)/d.
更大的表面积和更陡的浓度梯度可增加扩散速率，而更大的扩散距离（厚度）则会降低它。该定律可写成 R ∝ (A × ΔC)/d。
Adaptations of exchange surfaces maximise these factors: alveoli provide large surface area, thin epithelium, and constant ventilation maintains a steep O₂/CO₂ gradient. Villi in the gut operate similarly.
交换面适应性特征最大化这些因素：肺泡提供大表面积、薄上皮，持续通气维持陡峭的 O₂/CO₂ 梯度。肠绒毛以类似方式运作。
8. Surface Area to Volume Ratio | 表面积与体积比
As an organism or cell increases in size, its surface area to volume ratio (SA:V) decreases. This ratio is crucial for understanding the limitations of diffusion in meeting metabolic demands.
随着生物体或细胞体积增大，其表面积与体积比（SA:V）减小。该比值对于理解扩散满足代谢需求的局限性至关重要。
The relationship can be calculated using simple geometry. For a cube of side length L, surface area = 6L², volume = L³, so SA:V = 6/L. As L increases, the ratio falls. This explains why large organisms require specialised transport systems.
该关系可通过简单几何计算。对于边长为 L 的立方体，表面积 = 6L²，体积 = L³，因此 SA:V = 6/L。随着 L 增大，比值下降。这解释了为何大型生物需要专门的运输系统。
Small organisms like Amoeba rely on diffusion across their body surface because their high SA:V makes the process efficient. Larger organisms, such as mammals, have lungs, gills, and circulatory systems to compensate for their low SA:V.
像变形虫这样的小生物依靠体表扩散，因其高 SA:V 使得该过程高效。哺乳动物等大型生物具有肺、鳃和循环系统，以弥补其低 SA:V。
9. Respiratory Quotient (RQ) | 呼吸商
The respiratory quotient (RQ) indicates which respiratory substrate is being metabolised. It is calculated from the ratio of carbon dioxide produced to oxygen consumed during a given period.
呼吸商（RQ）表明正被代谢的呼吸底物。它由给定时间内产生的二氧化碳与消耗的氧气的比值计算得出。
RQ = CO₂ produced ÷ O₂ consumed
For carbohydrates, RQ = 1.0; for lipids it is about 0.7; for proteins it is typically around 0.9. Values deviating from these may indicate a mixed substrate use or anaerobic respiration.
对于碳水化合物，RQ = 1.0；对于脂质约为 0.7；对于蛋白质通常约为 0.9。偏离这些值可能表明混合底物利用或无氧呼吸。
Experimentally, RQ can be measured using a respirometer. Knowledge of RQ helps interpret metabolic states, such as hibernation (fat metabolism gives RQ ~0.7) or seed germination (carbohydrate use gives RQ ~1.0).
实验上可使用呼吸计测量 RQ。了解 RQ 有助于解释代谢状态，如冬眠（脂肪代谢给出 RQ ~0.7）或种子萌发（碳水化合物利用给出 RQ ~1.0）。
10. Net Primary Production (NPP) and Efficiency | 净初级生产力与生态效率
Gross primary production (GPP) is the total chemical energy fixed by photosynthesis in a given area and time. Plants use some of this energy for their own respiration (R), leaving net primary production (NPP) available to the next trophic level.
总初级生产力（GPP）是在给定面积和时间内光合作用固定的总化学能。植物将其中一些能量用于自身呼吸（R），剩余的可用于下一营养级的是净初级生产力（NPP）。
NPP = GPP − R
NPP represents the rate at which biomass is accumulated and is measured in units of energy per area per time (e.g., kJ m⁻² year⁻¹) or mass of dry biomass per area per time. It is the energy available to herbivores.
NPP 表示生物量积累的速率，以每单位面积每单位时间的能量（例如 kJ m⁻² year⁻¹）或单位面积单位时间的干生物量质量来衡量。这是可供植食动物利用的能量。
Ecological efficiency between trophic levels is often around 10%, but the actual percentage transfer can be calculated as (energy in new level ÷ energy in previous level) × 100. Low efficiency limits the length of food chains.
营养级之间的生态效率通常约为 10%，但实际传递百分比可计算为（新营养级中的能量 ÷ 上一营养级中的能量）× 100。低效率限制了食物链的长度。
11. Magnification Calculations | 放大倍数计算
In microscopy, magnification refers to how much larger an image is compared to the actual specimen. The relationship is straightforward and is often tested using photomicrographs or drawings with a scale bar.
在显微镜学中，放大倍数指图像比实际样本大多少。该关系简单明了，常使用带比例尺的显微照片或绘图进行考察。
Magnification = Image size ÷ Actual size
Ensure both measurements are in the same units. If the image size is measured in mm, convert the actual size (often given in μm) appropriately: 1 mm = 1000 μm. Rearranging gives Actual size = Image size ÷ Magnification.
确保两次测量使用相同单位。如果图像大小以 mm 测量，则适当转换实际大小（通常以 μm 表示）：1 mm = 1000 μm。移项可得 Actual size = Image size ÷ Magnification。
When given a scale bar, measure its length on the image, find the magnification, and then use it to determine specimen dimensions. This skill underpins accurate interpretation of cell and tissue structures.
当给出比例尺时，测量其在图像上的长度，求出放大倍数，然后用它确定样本尺寸。该技能是准确解释细胞和组织结构的基础。
12. Water Potential and Osmosis | 水势与渗透
Water potential (Ψ) is the measure of the tendency of water to move from one place to another by osmosis. It determines the direction of water movement across a partially permeable membrane.
水势（Ψ）是衡量水通过渗透从一处移动到另一处的趋势的量度。它决定了水穿过选择透过性膜的方向。
Ψ = Ψₛ + Ψₚ
Ψₛ is the solute potential (also called osmotic potential), which becomes more negative as solute concentration increases because solutes bind water molecules. Ψₚ is the pressure potential, resulting from the physical pressure exerted on the solution (e.g., turgor pressure in plant cells).
Ψₛ 是溶质势（也称渗透势），随着溶质浓度增加变得更加负，因为溶质结合水分子。Ψₚ 是压力势，由施加在溶液上的物理压力产生（例如植物细胞中的膨压）。
Pure water at atmospheric pressure has a water potential of zero. Water always moves from a region of higher (less negative) water potential to a region of lower (more negative) water potential. In plant cells, turgid cells have Ψₚ > 0, while plasmolyzed cells have Ψₚ approaching zero.
大气压下的纯水水势为零。水总是从水势较高（负值较小）的区域流向水势较低（负值较大）的区域。在植物细胞中，膨大细胞的 Ψₚ > 0，而质壁分离细胞的 Ψₚ 趋于零。
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