Mathematical modelling

Fractional occupancy of DNA targets with sensor, as a function of buffered [metal] ([M]_b), was modelled using Dynafit and the template script in ^{Supplementary Software}, using determined metal affinities (1/K ₁), DNA affinities (1/K ₃ and 1/K ₄) plus cellular abundance of each sensor (P) and DNA target (D) (Table ¹)⁴⁷. Where the standard deviations for the DNA affinities of Co(II)- and Zn(II)-bound proteins overlapped, average values generated by combining the data for both metals were used for 1/K ₄. These were: 3.6 × 10⁻⁸ M for Co(II)-Zur and Zn(II)₂-Zur, 1.4 × 10⁻⁵ M for Co(II)-RcnR and Zn(II)-RcnR, and 4.7 × 10⁻⁷ M for Co(II)-ZntR and Zn(II)-ZntR. To determine the amount of (P•M)•D for ZntR, the response for 'PD' was removed from the Dynafit script. A cell volume of 1 fl was used to calculate [P]_total and [D]_total from the number of protein assemblies per cell (i.e. dimers or tetramers) (Table 1) and target DNA binding sites per cell (assumed to be 1 copy per cell for RcnR and ZntR, 4 copies per cell for Zur due to additional gene targets², and 15 copies per cell for FrmR and FrmR^E64H due to the presence of a low copy number reporter plasmid)²⁹. Supplementary Data ¹ and ^{Supplementary Software} provide Dynafit script and Excel spreadsheet to enable the above calculations. DNA occupancy by each sensor was normalised from zero to one using the minimum and maximum DNA occupancy values. FrmR and FrmR^E64H were normalised to the same scale. The equilibrium concentration of free metal (i.e. [buffered metal]) was calculated using the equilibrium M + B ⇌ MB, where M = metal and B = buffer component. This was solved, via a quadratic equation, using initial concentrations of metal and buffer of 0.01 and 0.1 M, respectively, and each equilibrium constant (K ₅) value listed in Supplementary Data ¹. The following are step-by-step instructions to calculate fractional DNA occupancy with sensor as a function of buffered [metal] using Supplementary Data ¹ and ^{Supplementary Software}: (1) Input the number of protein assemblies (functional units i.e. dimer/tetramer), DNA binding sites and cell volume into the blue boxes within Supplementary Data ¹. (2) Input association constants K1, K3 and K4 into the green boxes within Supplementary Data ¹. (3) Open Dynafit and load script file ^{Supplementary Software}. Edit the following parameters 'M', 'D', 'B', 'Keq1-4' and responses 'PD' and 'PMD', using the values in the orange boxes from Supplementary Data ¹. (4) Adjust 'Keq5' to 1e-1. (5) Edit [set:P] with the contents of 'G6' and 'H6' from Supplementary Data ¹. (6) Edit location of Output file as appropriate. Currently C:/ (7) Run script. (8) Open output file 'data-model-t001-s001.txt' and obtain value for 'y(data)'. Input into cell 'N3' in Supplementary Data ¹. This gives the fractional occupancy of DNA at a buffered metal concentration of 5.12 × 10⁻³ M. (9) Repeat Dynafit simulation a further fourteen times. Each time, adjust 'Keq5' using values given in Supplementary Data ¹ column 'L', which will achieve corresponding buffered metal concentrations in column 'M'. Complete column 'N' as described in step 8. (10) Plot fractional DNA occupancy (column 'N') as a function of buffered metal concentration (column 'M') and adjust the x axis to logarithmic display. (11) Fractional DNA occupancy can then be normalised to a scale from zero to one if required.