### Classical formula for filtered samples

The classical formula can be found here or here

$$Pvol = DIC \cdot \frac{DPMvol}{DPMadded} \cdot 1.05 \cdot 1000$$

and plugging the value for $DPMadded$ (see below):

$$Pvol = DIC \cdot \frac{DPMvol}{SA \cdot V} \cdot 1.05 \cdot 1000$$

where:

$Pvol$ = Primary Productivity per unit volume in $mgC \cdot m^{-3} \cdot d^{-1}$

$DIC$ = Dissolved inorganic carbon in $mgC \cdot L^{-1}$

$DPMvol$ = DPM measured during 24 h for volume $V$ (filtered)

$DPMadded$ = $SA \cdot V$

$SA$ = Standard activity in 1 mL (DPM per mL)

$V$ = Volume of incubation (mL)

$1.05$ is to account for differential uptake between $^{14}$C and $^{12}$C

$1000$ is to go from from $L^{-1}$ to $m^{-3}$ since DIC is in $mgC \cdot L^{-1}$

### For cells sorted by flow cytometry

Let us define:

$DPMcells$ = DPM measured during 24 h for $N$ cells

$N$ = number of cells sorted

$C$ = cell concentration per mL

$Pcell$ = Primary productivity per cell ($fgC \cdot cell^{-1} \cdot h^{-1}$)

$Pcell$ can be computed from $Pvol$ by dividing it by the total number of cells in 1 $m^{-3}$:

$$Pcell = Pvol \cdot \frac{1}{C \cdot 10^{6} } \cdot \frac{1}{24} \cdot 10^{12}$$

$10^{12}$ is to convert from milligrams to femtograms

${24}$ is to convert from day to hours

$C \cdot 10^{6}$ is the number of cells per $m^{-3}$

Which simplifies to:

$$Pcell = Pvol \cdot \frac{1}{C} \cdot \frac{1}{24}\cdot 10^{6}$$

We now replace $Pvol$ by the formula provived at the top :

$$Pcell = DIC \cdot \frac{DPMvol}{SA \cdot V} \cdot 1.05 \cdot 1000 \cdot \frac{1}{C} \cdot \frac{1}{24} \cdot 10^{6}$$

Simplifies to :

$$Pcell = DIC \cdot \frac{DPMvol}{SA \cdot V \cdot C} \cdot 1.05 \cdot \frac{1}{24} \cdot 10^{9}$$

$DPMvol$ can be computed from the individual DPM per cell ($DPMcells/N$) as:

$$DPMvol = \frac{DPMcells \cdot C \cdot V }{N}$$

So that:

$$Pcell = DIC \cdot \frac{DPMcells \cdot C \cdot V}{SA \cdot V \cdot C \cdot N} \cdot 1.05 \cdot \frac{1}{24} \cdot 10^{9}$$

which simplifies finally to:

$$Pcell = DIC \cdot \frac{DPMcells}{SA \cdot N \cdot 24} \cdot 1.05 \cdot 10^{9}$$

With contributions from Adriana Lopes dos Santos (ASE, NTU) and Andres Gutierrez-Rodriguez (NIWA)