Background                                            7 nM                          1100 nM                         1 nM                        2.5 nM
  Concentration, $\bar{c}$                                                                                                            
  Concentration                                       10 nM/mm                       550 nM/mm                    $0.1$ nM/mm                  $0.5$ nM/mm
  Gradient, $\bar{g}$                                                                                                                 
  Percent Concentration                                 1.4%                           1.0%                           0.1%                         0.2%
  Difference, $\bar{p} = \bar{g}a/\bar{c}$                                                                                            
  Molecule Number                                        60                           53,000                          0.6                           3
  Difference, $\Delta\bar{n} = \bar{g}a^4$                                                                                            
  ------------------------------------------ --------------------------- --------------------------------- -------------------------- ------------------------------

Multicellular gradient sensing
------------------------------

In many cancer types, tumor cells invade the surrounding tissue in a collective manner [@cheung2013collective; @friedl2012classifying]. Cell collectives can sense shallower gradients than single cells [@ellison2016cell; @malet2015collective], both in terms of percent concentration changes and absolute molecule numbers (Table \[sense\_table\]). Indeed, groups of neurons respond to gradients equivalent to a difference of less than one molecule across an individual neuron’s growth cone [@rosoff2004new]. This raises the possibility that during the invasion process tumor cell collectives benefit from higher sensory precision than single tumor cells.

We can understand immediately from Eq. \[eq:g\] why a multicellular collective would have lower sensory error than a single cell: a collective is larger than a single cell. Therefore, the collective covers a larger portion of the concentration profile, which leads to a larger difference between the concentration measurements on either end, and a lower relative error. In terms of Eq. \[eq:g\], if we consider 