Table 2.

Summary of key biophysical scaling relationships. See main text for references and for *rare prokaryotic examples that defy the general trend. (We have assumed throughout: k_BT = 4.11 × 10⁻²¹ J = 4.11 pN nm, and μ/ρ = 10⁻² cm² s⁻¹ is the kinematic viscosity of water at room temperature). Symbols are defined in the text.

physical parameter	scaling	(typical) prokaryote	(typical) eukaryote	comments
Reynolds number	Re = ρUL/μ	Re = 10−5–10−3 (L = 1–10 μm, U = 10–100 μm s−1)	Re = 10−3–1 (L = 10–1000 μm, U = 10–1000 μm s−1)	unicellular eukaryotes and prokaryotes are viscosity dominated
Péclet number (fix D = 103 μm2 s−1 for small molecule)	Pe = UL/D	Pe = 10−2–1 (L, U as above)	Pe = 1–103 (L, U as above)	prokaryotes are diffusion-limited and cannot reach* Pe > 1
diffusion constant (passive)	D0 = kBT/(6πμa)	D0 = 0.22 μm2 s−1 (a = 1 μm)	D0 = 0.02 μm2 s−1 (a = 10 μm)	eukaryotes 10× less susceptible to linear diffusion
rotational diffusion (passive)	Drot = kBT/(8πμa3)	Drot = 0.16 rad2 s−1 (a = 1 μm)	Drot = 1.6 × 10−4 rad2 s−1 (a = 10 μm)	eukaryotes 1000× less susceptible to rotational diffusion
effective diffusion (active, depends on motility strategy)	Da = v2τ/3 (τ = free-flight time, v = speed of runs)	Da ≈ 133 μm2 s−1 (v ≈ 20 μm s−1, τ ≈ 1 s)	Da ≈ 3.3 × 10−2 cm2 s−1 (v ≈ 100 μm s−1, τ ≈ 10 s)	empirical: E. coli: [309] D ≈ 10–100 μm2 s−1; C. reinhardtii: [310] D ≈ 10−3 cm2 s−1; P. caudatum: [311] D ≈ 10−2 cm2 s−1
relevant for stochastic navigation	expected angular deviation in time τ θrms = √(2Drτ)	θrms = 32° (for τ = 1 s)	θrms = 1° (for τ = 1 s)	prokaryotes are severely limited by rotational diffusion (cannot steer)
relevant for spatial sensing	${\displaystyle \frac{\text{SN}{\text{R}}_{\mathrm{s}}}{\text{SN}{\text{R}}_{\mathrm{t}}}\sim \frac{a}{v\tau }\hspace{0.17em}}$	too small*	useful strategy for large and slow-moving cells	common strategy for amoeboid eukaryotes
relevant for helical klinotaxis	${\mathrm{SNR}}_{\mathrm{klinotaxis}}\sim {\displaystyle \frac{R^2}{\omega }}$	too small*	useful strategy to reorient toward vectorial cues	common strategy for free-swimming eukaryotes
sensitivity to mechanical stimuli (membrane tension)	ratio of channel opening and closing probabilities ${\displaystyle \frac{P_0}{P_{\mathrm{c}}}=\exp \left(-\frac{\mathrm{\Delta }E}{k_{\mathrm{B}}T}\right)}$ ΔE = work done to open channel (approx. sensitivity)	bacterial MscS and MscL channels (osmotic nanovalves): [301] 5 approximately 10 mN m−1	hair cells, single-channel gating stiffness: [308] approximately 1 mN m−1 Piezo1: [312] approximately 1.4 mN m−1	1 kBT is 4 nm2 change in area under tension of 1 mN m−1 lytic tension of pure lipid bilayer approximately 20 mN m−1