Biology formula sheet

Biology All Biology tools

• Cell Doubling Time Calculator

  Doubling time and growth rate from two counts, or project forward.

  Exponential growth kinetics

  $$T_d = \frac{t \ln 2}{\ln (N/N_0)}$$

• Centrifuge RCF Calculator

  Convert RPM to × g and back for your rotor’s radius.

  Definition of relative centrifugal force

  $$\mathrm{RCF} = 1.118 \times 10^{-5} \, r \, \mathrm{RPM}^{2}\quad(r\text{ in cm})$$

• Nucleic Acid Quantification Calculator

  Turn an A260 reading into ng/µL of DNA or RNA, and get the volume needed for a target mass.

  Beer’s law (1852) with standard A260 conversion factors

  $$c = A_{260} \times \varepsilon \times d$$

• Action Potential Simulator

  Hodgkin-Huxley, with threshold and refractoriness emerging.

  Hodgkin and Huxley (1952)

  $$C_m\frac{dV}{dt} = I - I_{Na} - I_K - I_L$$

• Oxygen Dissociation Curve Simulator

  The sigmoid curve, and what moves it left or right.

  Hill (1910), with Severinghaus (1979) standard curve

  $$S = \frac{(P/P_{50})^{n}}{1 + (P/P_{50})^{n}}$$

• Enzyme Kinetics Simulator

  Km and Vmax, and how each inhibitor changes them.

  Michaelis and Menten (1913), Briggs and Haldane (1925)

  $$v = \frac{V_{max}[S]}{K_m + [S]}$$

• Acid Base Interpreter

  A blood gas worked through, one reasoned step at a time.

  Henderson-Hasselbalch, with Albert, Dell and Winters (1967)

  $$\mathrm{pH} = 6.1 + \log_{10}\!\left(\frac{[\mathrm{HCO_3^-}]}{0.03\,P_{\mathrm{CO_2}}}\right)$$

• Winters’ Formula Calculator

  Expected pCO2 in a metabolic acidosis, and what a mismatch means.

  Albert, Dell and Winters (1967)

  $$\text{expected } P_{\mathrm{CO_2}} = 1.5\,[\mathrm{HCO_3^-}] + 8 \pm 2$$

• eGFR and Creatinine Clearance Calculator

  Three estimates of kidney function, and where they diverge.

  Cockcroft and Gault (1976), MDRD Levey (2006), CKD-EPI Inker (2021)

  $$C_{Cr} = \frac{(140 - \text{age})\times \text{mass}}{72 \times S_{Cr}}$$

• Mean Arterial Pressure Calculator

  MAP and pulse pressure from a blood pressure.

  Diastolic pressure plus one third of pulse pressure

  $$\text{MAP} = P_{dia} + \frac{P_{sys} - P_{dia}}{3}$$

• Target Heart Rate Calculator

  Training heart rate zones from age and resting rate.

  Karvonen (1957), Tanaka, Monahan and Seals (2001), ACSM bands (Garber 2011)

  $$\text{THR} = \text{HR}_{rest} + \text{intensity} \times (\text{HR}_{max} - \text{HR}_{rest}),\quad \text{HR}_{max} = 208 - 0.7 \times \text{age}$$

• Albumin Corrected Calcium Calculator

  Total calcium corrected for albumin, both units.

  Payne (1973), albumin-adjusted calcium

  $$\text{Ca}_{corr} = \text{Ca} + 0.8\,(4.0 - \text{albumin})$$

• Serum Osmolality and Osmolar Gap Calculator

  Calculated osmolality, the gap, and tonicity.

  Smithline and Gardner (1976), calculated osmolality

  $$\text{Osm} = 2\,[\text{Na}^+] + \frac{[\text{glucose}]}{18} + \frac{[\text{BUN}]}{2.8}$$

• Anion Gap Calculator

  Anion gap, albumin correction and delta ratio.

  Figge, Jabor, Kazda and Fencl (1998), albumin-corrected anion gap

  $$\text{AG}_{corr} = [\text{Na}^+] - [\text{Cl}^-] - [\text{HCO}_3^-] + 2.5\,(4.0 - \text{albumin})$$

• Fractional Excretion of Sodium (FENa) Calculator

  FENa and FEUrea from a spot urine and a blood test.

  FENa Espinel (1976), FEUrea Carvounis (2002)

  $$\text{FE}_{\text{Na}} = \frac{U_{\text{Na}} \times S_{\text{Cr}}}{S_{\text{Na}} \times U_{\text{Cr}}} \times 100$$

• Corrected Sodium and Free Water Deficit Calculator

  Glucose-corrected sodium and the free water deficit.

  Katz (1973) and Hillier et al. (1999); body water fractions, Adrogué and Madias (2000)

  $$\text{Na}_{corr} = \text{Na} + 1.6 \times \frac{\text{glucose} - 100}{100}$$

• A-a Gradient and Alveolar Gas Equation Calculator

  Alveolar oxygen, the A-a gradient and P/F ratio.

  Alveolar gas equation, Riley and Cournand (1949)

  $$P_AO_2 = F_iO_2\,(P_{atm} - P_{H_2O}) - \frac{P_aCO_2}{R}$$

• Arterial Oxygen Content Calculator (CaO2 and DO2)

  CaO2 and DO2 from hemoglobin and a blood gas.

  Oxygen content equation, Hüfner factor 1.34

  $$C_aO_2 = 1.34 \cdot \text{Hb} \cdot S_aO_2 + 0.003 \cdot P_aO_2$$

• QTc Calculator

  Four QT corrections at once, with thresholds.

  Bazett (1920), Fridericia (1920), Hodges (1983), Sagie (1992)

  $$QTc = \frac{QT}{\sqrt{RR}}, \quad RR = \frac{60}{HR}$$

• Body Surface Area Calculator

  Four BSA formulas, and how far apart they land.

  Du Bois (1916), Gehan and George (1970), Haycock (1978), Mosteller (1987)

  $$\text{BSA} = \sqrt{\frac{h_{cm} \times w_{kg}}{3600}}$$

• BMI Calculator

  Adult BMI, category, BMI prime and healthy weight range.

  Keys et al. (1972), after Quetelet; WHO (2000) and CDC adult categories

  $$\text{BMI} = \frac{w_{\text{kg}}}{(h_{\text{m}})^{2}} \approx 703 \times \frac{w_{\text{lb}}}{(h_{\text{in}})^{2}}$$

• Ideal Body Weight Calculator

  IBW, adjusted weight, PBW and tidal volumes.

  Devine (1974), Bauer et al. (1983), ARDS Network (2000)

  $$\text{IBW} = B + 2.3\,(h_{\text{in}} - 60),\quad B = 50\text{ kg (men)},\ 45.5\text{ kg (women)}$$

• BMR and TDEE Calculator

  BMR by three equations, and TDEE from activity.

  Mifflin et al. (1990), Roza and Shizgal (1984), Cunningham (1991), FAO/WHO/UNU (2004)

  $$\text{BMR} = 10\,w_{\text{kg}} + 6.25\,h_{\text{cm}} - 5\,a + s,\quad s = +5\text{ (men)},\ -161\text{ (women)}$$

• Body Fat Calculator

  Navy tape method body fat, with the ACE category.

  Hodgdon and Beckett (1984), Siri (1961)

  $$\%\,\text{fat}_{\text{men}} = \frac{495}{1.0324 - 0.19077\log_{10}(w - n) + 0.15456\log_{10} h} - 450$$

• Calories Burned Calculator

  Calories burned by activity, weight and time.

  Herrmann et al. (2024), 2024 Adult Compendium of Physical Activities

  $$\text{kcal} = \text{MET} \times m_{\text{kg}} \times t_{\text{h}}$$

• IV Drip Rate Calculator

  Drops per minute from volume, time and drop factor.

  Volume times drop factor over time, by dimensional analysis

  $$\text{gtt/min} = \frac{\text{volume (mL)} \times \text{drop factor (gtt/mL)}}{\text{time (min)}}$$

• IV Infusion Rate Calculator

  Pump rate in mL/hr from volume and time.

  Flow rate as volume delivered per unit time

  $$\text{mL/h} = \frac{\text{volume (mL)}}{\text{time (h)}}$$

• mcg/kg/min to mL/hr Calculator

  Pump rate in mL/hr from a dose in mcg/kg/min.

  Dose rate times weight over concentration, by dimensional analysis

  $$\text{mL/h} = \frac{\text{mcg/kg/min} \times \text{kg} \times 60}{\text{mcg/mL}}$$

• Weight-Based Dosage Calculator

  Dose in mg and volume in mL from mg/kg and weight.

  Dose per kilogram times weight, over concentration

  $$\text{mL} = \frac{\text{dose (mg/kg)} \times \text{weight (kg)}}{\text{concentration (mg/mL)}}$$

• DNA Double Helix Explorer

  The grooves, and the strand offset that creates them.

  Watson and Crick (1953), with Franklin and Gosling (1953)

  $$\text{bp per turn} = \frac{360^\circ}{\text{twist}}, \quad \text{pitch} = \text{rise} \times \text{bp per turn}$$

• Nephron Explorer

  The loop, the gradient it builds, and what ADH can do with it.

  Countercurrent multiplication, Kuhn and Ryffel (1942)

  $$\dot{V}_{\text{urine}} = \frac{\dot{n}_{\text{osm}}}{U_{\text{osm}}}, \quad C_{\mathrm{H_2O}} = \dot{V} - \frac{\dot{V}\,U_{\text{osm}}}{P_{\text{osm}}}$$

• Resting Membrane Potential Simulator

  Nernst per ion, Goldman for the membrane.

  Nernst (1889), with Goldman (1943) and Hodgkin and Katz (1949)

  $$V_m = \frac{RT}{F}\ln\frac{P_K[\text{K}^+]_o + P_{Na}[\text{Na}^+]_o + P_{Cl}[\text{Cl}^-]_i}{P_K[\text{K}^+]_i + P_{Na}[\text{Na}^+]_i + P_{Cl}[\text{Cl}^-]_o}$$

• Primer Tm Calculator

  Primer Tm from the sequence and buffer by nearest neighbours, with the Wallace and GC rules beside it.

  SantaLucia (1998), with the Mg²⁺ term of von Ahsen et al. (2001)

  $$T_m = \frac{\Delta H^\circ}{\Delta S^\circ + 0.368\,(N - 1)\ln[\mathrm{Na^+}] + R\ln(C_T/4)}\quad(\Delta H^\circ\text{ in cal/mol, }[\mathrm{Na^+}]\text{ and }C_T\text{ in mol/L, }T_m\text{ in K})$$

• Ligation Calculator

  Nanograms of insert for any insert:vector molar ratio, from vector mass and both lengths.

  Insert to vector molar ratio, as given in Promega’s pGEM-T manual (TM042)

  $$m_{\text{insert}} = R \times m_{\text{vector}} \times \frac{L_{\text{insert}}}{L_{\text{vector}}}$$

• Microscope Magnification Calculator

  Magnification, actual size, field of view and cell size for a light microscope.

  Definition of magnification, AQA GCSE Biology specification 8461, section 4.1.1.5

  $$M = \frac{\text{image size}}{\text{actual size}}$$

• Punnett Square Calculator

  Genotype and phenotype ratios for crosses of up to three genes, with the square drawn.

  Mendel (1866), segregation and independent assortment

  $$P(AaBb) = P(Aa) \times P(Bb)$$

• Reverse Complement Calculator

  Reverse complement, complement or reverse of DNA or RNA, with IUPAC codes, length and GC content.

  Watson and Crick (1953), with the NC-IUB codes of Cornish-Bowden (1985)

  $$r_i = c(s_{n+1-i}), \quad \mathrm{GC} = \frac{G + C + S}{A + C + G + T + U + S + W} \times 100\%$$

• SIR Epidemic Model Simulator

  SIR and SEIR epidemics with R₀, herd immunity and vaccination.

  Kermack and McKendrick (1927)

  $$\frac{dS}{dt} = -\beta\frac{SI}{N}, \quad \frac{dI}{dt} = \beta\frac{SI}{N} - \gamma I$$

• Predator-Prey Simulator

  Lotka-Volterra cycles over time and in the phase plane, with the equilibrium and exact period.

  Lotka (1925) and Volterra (1926)

  $$\frac{dx}{dt} = \alpha x - \beta xy,\quad \frac{dy}{dt} = \delta xy - \gamma y$$

• Photosynthesis Rate Simulator

  Light, CO₂ and temperature against the rate, with the limiting factor named and pondweed bubbles counted.

  Farquhar, von Caemmerer and Berry (1980)

  $$A = \min(A_c, A_j) - R_d$$

• Cardiac Cycle Simulator

  Pressures, volume, valves and ECG through one heartbeat.

  Time-varying elastance, Suga and Sagawa (1974)

  $$P_{\text{LV}} = e(t)\,E_{\text{es}}(V - V_0) + \left(1 - e(t)\right)P_0\,e^{\beta V}$$

• DNA to Protein Translator

  Paste DNA or mRNA and read off the protein, codon by codon, in every reading frame.

  Crick, Barnett, Brenner and Watts-Tobin (1961), with NCBI translation table 1

  $$n_{\text{aa}} = \frac{L_{\text{ORF}}}{3} - 1, \quad 4^{3} = 64 \text{ codons}$$

• Diffusion and Osmosis Simulator

  Diffusion or osmosis across one membrane, counted against Fick’s law.

  Fick (1855) and van ’t Hoff (1887)

  $$J = -D\frac{dc}{dx}, \quad \Pi = cRT$$

• Hardy-Weinberg Calculator

  Allele and genotype frequencies from one known value or from counts, with a chi-square test of equilibrium.

  Hardy (1908) and Weinberg (1908)

  $$p^{2} + 2pq + q^{2} = 1, \quad p + q = 1$$

• Cell Structure Explorer

  Labelled animal and plant cells, with what every organelle is, does and measures.

  Surface area and volume of a sphere, as in OpenStax Biology 2e (2018), section 4.2

  $$\frac{A}{V} = \frac{4\pi r^{2}}{\tfrac{4}{3}\pi r^{3}} = \frac{3}{r}$$

• Mitosis and Meiosis Explorer

  Mitosis and meiosis stage by stage, with chromosome and chromatid counts for your 2n.

  Sutton (1903), The Chromosomes in Heredity

  $$N_{\text{gametes}} = 2^{n}, \quad N_{\text{zygotes}} = 2^{n} \times 2^{n} = 4^{n}$$

• Protein Synthesis Simulator

  Type a gene and watch RNA polymerase, a ribosome and tRNAs build its protein, step by step.

  Elongation rates from Milo and Phillips (2015), Cell Biology by the Numbers

  $$t = \frac{L}{v}$$

• Water Potential Calculator

  Solute and water potential for a cell and the solution around it, the way water moves, and the state the cell ends in.

  van ’t Hoff (1887) and the AP Biology equation sheet

  $$\psi = \psi_{s} + \psi_{p}, \quad \psi_{s} = -iCRT$$

• Mark-Recapture Population Calculator

  Population size from marked, caught and recaptured counts by Lincoln-Petersen and Chapman, with a 95% interval.

  Petersen (1896), Lincoln (1930) and Chapman (1951)

  $$N = \frac{MC}{R}, \quad N_{C} = \frac{(M+1)(C+1)}{R+1} - 1$$

• Simpson’s Diversity Index Calculator

  Simpson’s D, 1 − D and 1/D from species counts, with Shannon’s index, evenness and a second habitat to compare.

  Simpson (1949), Measurement of diversity, Nature 163: 688; Shannon (1948)

  $$D = \frac{\sum n(n-1)}{N(N-1)}, \quad H' = -\sum p \ln p$$

• qPCR and ΔΔCt Simulator

  Four Ct values to a fold change by 2^−ΔΔCt or Pfaffl, with amplification curves, a threshold and a standard curve.

  Livak and Schmittgen (2001); Pfaffl (2001) for unequal efficiencies

  $$\text{fold change} = 2^{-\Delta\Delta C_t},\quad \Delta\Delta C_t = (C_{t,\text{target}} - C_{t,\text{ref}})_{\text{treated}} - (C_{t,\text{target}} - C_{t,\text{ref}})_{\text{control}}$$

• Muscle Twitch and Tetanus Simulator

  Fire a muscle at any frequency and watch single twitches sum into unfused and then fused tetanus.

  Milner-Brown, Stein and Yemm (1973) twitch, summed as in Fuglevand, Winter and Patla (1993)

  $$A(t) = \sum_k \frac{t - t_k}{T}\, e^{1 - (t - t_k)/T}, \quad F = F_{\text{max}}\left[1 - (1 - \rho)^{A}\right]$$

• Pharmacokinetics Dosing Simulator

  Build repeated doses up to steady state and read the peak, trough, average, accumulation factor and loading dose.

  Gibaldi and Perrier, Pharmacokinetics (1982); Rowland and Tozer (2011)

  $$C_{\mathrm{ss}} = \frac{F \cdot D}{\mathrm{CL} \cdot \tau}$$

• ECG (EKG) Rhythm and Heart Block Simulator

  A sweeping lead II strip and ladder diagram for sinus rhythm, heart blocks, AF, flutter and ectopic beats.

  Kligfield et al. (2007) ECG recording standards; Kusumoto et al. (2018) ACC/AHA/HRS guideline

  $$\text{HR} = \frac{60}{RR} = \frac{1500}{RR_{\text{mm}}}$$

• Bacterial Unknown Identification Lab

  Run standard tests on 15 teaching bacteria, or enter your own results, and watch a dichotomous key narrow the field.

  Reference reactions from Farmer et al., J. Clin. Microbiol. 21:46 (1985), and Bergey’s Manual of Systematic Bacteriology

  $$o \text{ fits} \iff x(t) \in R(o, t) \text{ for every test } t \text{ run}$$

• Gel Electrophoresis Simulator

  Run a virtual agarose gel with a ladder and digests, and size an unknown band from the semi-log standard curve.

  Southern (1979); Helling, Goodman and Boyer (1974); Stellwagen, Gelfi and Righetti (1997)

  $$d = \frac{\mu_0 E t}{1 + L / L_{1/2}}, \quad \log_{10} L \approx a - b\,d$$

• Spirometry and Flow-Volume Loop Simulator

  Watch normal, obstructive, restrictive and upper airway flow-volume loops form, with FEV1/FVC read out.

  ATS/ERS interpretive strategies: Pellegrino et al. (2005), Stanojevic et al. (2022)

  $$\frac{\mathrm{FEV_1}}{\mathrm{FVC}} < \mathrm{LLN}$$

• Amino Acid 3D Explorer

  All 20 amino acids in 3D, grouped by side chain, their charges changing as you move the pH.

  Henderson-Hasselbalch per ionisable group; pKa values from Nelson and Cox, Lehninger Principles of Biochemistry, Table 3-1

  $$q(\mathrm{pH}) = \sum_{\text{basic}} \frac{1}{1 + 10^{\mathrm{pH} - \mathrm{p}K_a}} - \sum_{\text{acidic}} \frac{1}{1 + 10^{\mathrm{p}K_a - \mathrm{pH}}}$$

• Blind Spot Mapper

  Watch a dot vanish, as Mariotte’s paper did in 1668, then map your own blind spot in degrees and see it as the optic disc on the retina.

  Mariotte (1668) for the blind spot; the virtual chinrest of Li, Joo, Yeatman and Reinecke, Sci. Rep. 10:904 (2020)

  $$\theta = \arctan\frac{x}{d}$$

• Blood Type and X-Linked Punnett Square Calculator

  A child’s possible ABO and Rh types from the parents’, and X-linked crosses split by sons and daughters.

  Bernstein (1924), ABO blood groups as three alleles at one locus

  $$P(\text{AO} \mid \text{A}) = \frac{2pr}{p^{2} + 2pr}$$

• Caffeine Level Simulator

  Your drinks on a timeline, and the caffeine they leave in you hour by hour.

  Bateman (1910), one-compartment oral absorption

  $$A(t) = \sum_i D_i\,\frac{k_a}{k_a - k}\left(e^{-k(t - t_i)} - e^{-k_a(t - t_i)}\right)$$

• Color Blindness Simulator

  Any image as protan, deutan or tritan eyes see it, at any severity, with the cone curves that explain why.

  Machado, Oliveira and Fernandes (2009), with Brettel, Viénot and Mollon (1997)

  $$c' = \Phi(\text{type}, s) \cdot c$$

• Hearing Range Test and Hearing Loss Simulator

  Find the highest pitch you can hear, see each pitch as a place on the cochlea, and hear music through a hearing loss.

  Greenwood (1990), J Acoust Soc Am 87(6):2592

  $$f = A\left(10^{a x} - k\right)$$

• Menstrual Cycle Hormone Simulator

  FSH, LH, estrogen and progesterone through the cycle, with the follicle and the lining redrawn each day.

  WHO Task Force (1980), Am J Obstet Gynecol 138:383

  $$t_{\text{ov}} = t_{\text{LH peak}} + 16.5\,\text{h}$$

• PCR Simulator

  Watch PCR cycle by cycle, find where primers bind on a typed template, and run the product on a gel.

  Mullis and Faloona (1987); SantaLucia (1998) for primer binding

  $$N_c = N_0\,(1 + E)^{c},\quad E = \sqrt{\theta_F\,\theta_R},\quad \text{exact-length duplexes} = 2^{n} - 2n$$

• Reflex Arc Simulator

  Tap the knee and follow the reflex arc from spindle to spinal cord to muscle, timed to the millisecond.

  Definition of reflex latency: conduction time plus synaptic and junctional delays

  $$t_{\text{EMG}} = t_r + \sum_i \frac{L_i}{v_i} + n\,\delta_{\text{syn}} + \tau + \delta_m$$

• Sleep Cycle and Hypnogram Visualizer

  A typical night’s sleep stages from your bedtime to your alarm, with deep sleep early, REM late and the windows where waking is easiest.

  Borbély (1982), two-process model, with time constants from Daan, Beersma and Borbély (1984)

  $$S(t) = S_0\,e^{-t/\tau_d},\quad \tau_d = 4.2\ \text{h}$$

• One Rep Max Calculator

  1RM from weight and reps, with a percentage chart.

  Epley (1985), Brzycki (1993), Lombardi (1989)

  $$\text{1RM}_{\text{E}} = w\left(1 + \frac{r}{30}\right),\quad \text{1RM}_{\text{B}} = \frac{w}{1.0278 - 0.0278\,r},\quad \text{1RM}_{\text{L}} = w\,r^{0.10}$$

• Reaction Time Ruler-Drop Experiment

  Catch a falling ruler: free fall turns the catch distance into your reaction time, run after run.

  Galileo, Two New Sciences (1638)

  $$t = \sqrt{\frac{2d}{g}}$$