Table 1.

Summary of typical biological parameters and estimated integration times to reach accuracy of 10% in measuring ligand concentration

Parameter	Value	Reference
D2	0.01 to 0.1 μm2/s	(22,23)
kendo	0.001 to 0.01 1/s	(24–27)
C	1 to 100 μm−2	(22,28–30)
S	0.1 to 5 μm	Small range corresponds to filopodia, large range corresponds to epithelial contacts.
${\tau }_{min}\left(toreach\frac{\delta c}{c}=0.1\right)$	0.91 s∗	Calculated based on Eq. 12
${\tau }_{max}\left(toreach\frac{\delta c}{c}=0.1\right)$	3.22 h∗∗	Calculated based on Eq. 13
${\tau }_{3D,typical}\left(toreach\frac{\delta c}{c}=0.1\right)$	0.03 s ∗∗∗	Calculated based on (4)

All quantities were calculated using the limit of large number of receptors.

^∗τ_min ∼ 0.91 s is calculated by numerically solving the equation, ${\left(\delta c/\overline{c}\right)}^2=1/\pi m{D}_2\tau \overline{c}\left[\ln \left({\lambda }^{\ast }/a\right)+{\sum }_{i=1}^{m-1}{K}_0\left(\left(2s/{\lambda }^{\ast }\right)\sin \left(\pi i/m\right)\right)\right]$, and assuming the following parameters: k_endo = 10⁻² 1/s, D₂=10⁻¹ μm ²/s, $\overline{c}$ = 100 1/μm ², $\delta c/\overline{c}=0.1$, s = 5 μm, and m = 20.

^∗∗${\text{τ}}_{\text{max}}$ ∼ 3.22 h is calculated by solving Eq. 13 with $\lambda =\sqrt{D_2/k_{endo}}$, and assuming the following parameters: k_endo = 10⁻³ 1/s, D₂ = 10⁻² μm ²/s, $\overline{c}$ = 1 1/μm ², $\delta c/\overline{c}=0.1$, s = 0.1 μm, and m = 20. We note that in this limit solving the exact formula (Eq. 12) gives the same result.

^∗∗∗${\text{τ}}_{3\text{D},\text{typical}}$ ∼ 0.03 s is calculated using ${\tau }_{3D}\approx 1/{2\pi D_3\overline{c}a\left(\delta c/\overline{c}\right)}^2$ (4) by assuming the following parameters: D₃ = 10 μm ²/s, $\overline{c}=100nM$, a = 1 μm.