Units, Numbers, and Energy in Biological Systems: Foundations for Microbiology

Units, Numbers, Energy

Numbers and Units

Understanding numbers and units is fundamental in microbiology, as quantitative measurements are essential for describing biological phenomena. Accurate use of units allows for clear communication and reproducibility in scientific research.

• Metric Prefixes: Prefixes such as kilo (k), milli (m), micro (μ), nano (n), and pico (p) are used to express quantities over a wide range of magnitudes. For example, 1 mg = 10-3 g, 1 μg = 10-6 g.

• Conversion between Mass and Molar Units: The mole is a standard unit for amount of substance, defined as 6.02 × 1023 molecules. Molarity (M) is moles per liter, and conversions often use molecular weight.

• Typical Concentrations and Sizes: Biological systems span a wide range of concentrations and sizes, from nanometers (nm) for molecules to millimeters (mm) for cells.

• Length, Area, and Volume: Calculations of area and volume are important for understanding physiological processes, such as diffusion and metabolic rates.

"When you can measure what you are speaking about, and express it in numbers, you know something about it. When you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind." – Lord Kelvin

Logarithms and Exponentials

Logarithmic and exponential functions are widely used in biology to describe growth, decay, and concentration changes.

• Logarithms: The logarithm of a number is the exponent to which the base must be raised to produce that number. For example, log10(100) = 2.

• Exponentials: Exponential growth or decay is described by equations such as $y = 10^x$.

• Applications: Logarithmic scales are used for pH, concentrations, and plotting biological data that span several orders of magnitude.

x	10x	log(x)
-2	0.01	-2
-1	0.1	-1
0	1	0
1	10	1
2	100	2

Log Scale Graphs

Logarithmic scales are used to visualize data that cover a wide range of values, such as metabolic rates versus body mass.

• Distance on Axis: On a log scale, equal distances represent equal ratios, not equal differences.

• Example: Mass-specific metabolic rate plotted against log(mass) for various animals.

Distance	Mass
0.5	3.16
1	10
1.7	50
2	100
3	1000

Metric Prefixes and Canceling Prefixes

Metric prefixes simplify the expression of very large or very small numbers.

• Common Prefixes: M = 106, k = 103, m = 10-3, μ = 10-6, n = 10-9, p = 10-12

• Canceling Prefixes: 5 mg/mL = 5 g/L = 5 μg/μL = 5000 pg/μL

• Scientific Notation: Use scientific notation for clarity (e.g., 10-12 grams instead of 0.000000000001 grams).

Area and Volume Calculations

Calculating area and volume is essential for understanding physiological and geometric properties in biology.

• Cube Example: A cube with side 100 cm has a total surface area of $6 imes (100 ext{ cm} imes 100 ext{ cm}) = 6 imes 10^4 ext{ cm}^2$.

• Scaling: Splitting into smaller cubes changes the total area but not the total volume.

Area/Volume Relationships

The ratio of surface area to volume affects many biological processes, such as diffusion and metabolic rates.

• Formulas: $A ext{ (area)} hicksim d^2$, $V ext{ (volume)} hicksim d^3$

• Implications: As size increases, volume grows faster than area, affecting transport and metabolic rates.

• Examples: Travel time of nerve impulse $hicksim d$, rate of heat/mass exchange $hicksim d^2$, rate of metabolic processes $hicksim d^3$

On Being the Right Size

Body size influences physiological capabilities, such as strength and metabolic rate.

• Example: Olympic weightlifting records show that relative strength decreases as body weight increases.

Concentrations and Molarity

Concentration is a key concept in microbiology, used to describe the amount of solute in a solution.

• Mole: 1 mole = $6.02 imes 10^{23}$ molecules

• Molarity (M): $1 ext{ M} = 1 ext{ mole/L}$

• Calculation Example: For a 20 g/L solution with molecular weight 200, $20/200 = 0.1 ext{ M}$

Concentration Example: Proteins in Cytoplasm

• Given: 200 g/L protein, average molecular weight 50,000

• Calculation: $200/50 = 4 ext{ mM}$

Molar Concentration and Average Distance Between Molecules

The average distance between molecules in solution depends on concentration.

• Formula: $d_{av} ( ext{nm}) = rac{1.2}{ oot{3}{C}}$

• Trend: Higher concentration leads to shorter average distance between molecules.

Percent Concentrations

Percent concentration (w/v) is commonly used in clinical and laboratory settings.

• Definition: 1% = 1 g/100 mL

• Example: 0.9% NaCl = 0.9 g/100 mL = 9 g/L = 154 mM

pH and Hydrogen Ion Concentration

pH is a logarithmic measure of hydrogen ion concentration, crucial for understanding biochemical environments.

• Formula: $pH = - ext{log}[H^+]$

• Examples: Bleach (pH 13), household ammonia (pH 11), pure water (pH 7), lemon juice (pH 2), gastric acid (pH 1), acid mine water (pH -3.6)

Examples of Concentrations and Sizes

Concentration	Value
Pure water	~55 M
NaCl in saline	154 mM
Biologically active compounds	10-7 - 10-9 M
1 molecule in a bacterial cell	~1 nM

Size	Value
Smallest visible by eye	0.1 mm
Bacteria	1-10 μm
Water molecule	0.3 nm
Proteins	2-10 nm
Average distance at 1 M	1 nm
Average distance at 1 mM	120 nm

Equilibrium Binding Curve and Affinity

Binding curves describe how the fraction of occupied binding sites depends on ligand concentration and affinity.

• High Affinity: Steep curve, saturation at low ligand concentration.

• Low Affinity: Gradual curve, saturation at higher ligand concentration.

Signaling by Concentration

Cellular signaling often depends on the concentration of signaling molecules, such as calcium ions.

• Fraction of Active Protein: Depends on the dissociation constant ($K_d$) and ligand concentration.

• Example: $K_d = 0.01 ext{ μM}$, $K_d = 1 ext{ μM}$, $K_d = 100 ext{ μM}$

Binding Curve in Linear and Logarithmic Scales

The same binding curve can appear different when plotted on linear versus logarithmic axes, but the underlying relationship remains unchanged.

Affinity and Specificity

Affinity refers to the strength of binding between a ligand and its target, while specificity describes the preference for one ligand over others.

• Example: Isoproterenol, epinephrine, and norepinephrine binding to β1 and α receptors with different affinities.

Effect of a Drug on Blood Pressure

Pharmacological effects are often quantified by dose-response curves, which resemble ligand binding curves.

• Example: Dexmedetomidine decreases blood pressure in rabbits, with a steep initial response and plateau at higher doses.

Effect of Scorpion Venom on Mice

Toxicology studies use dose-response curves to determine lethal and effective doses.

• Median Lethal Dose (LD50): The dose at which 50% of subjects die (e.g., LD50 = 1.075 μg/g for scorpion venom).

• Median Effective Dose (ED50): The dose of antivenom that protects 50% of subjects (e.g., ED50 = 12.54 μL).

Additional info: These foundational concepts are essential for understanding microbial physiology, biochemistry, and experimental design in microbiology. Mastery of units, concentrations, and quantitative relationships enables students to interpret data and conduct meaningful research.