Phytoplankton Physiology

The various resources and processes are summarized in the schematic below, and further detail is provided in the following sub-sections.

State Variables

Each phytoplankton individual is characterized using eight state variables (see table below).

Photosynthesis

The parameterization for photosynthesis is formulated as

\[PS=PC \cdot Bm\]

\[PC=PC_{max}\cdot (1-e^{\frac{-\alpha\Phi \cdot I\cdot Chl}{PC_{max}\cdot Bm}})\]

\[PC_{max}= PCmax \cdot Sz^{PC_b}\]

where $PS$ is cell-specific light limited photosynthesis rate ($mmol~C~cell^{-1}~s^{-1}$). $PC$ is carbon-specific light limited photosynthesis rate ($mmol~C~(mmol~C)^{-1}~s^{-1}$). $PC_{max}$ is light-saturated photosynthesis rate ($mmol~C~(mmol~C)^{-1}~s^{-1})$ and is scaled by a power-law relationship of cell size ($Sz$), $I$ is the flux of photosynthetically active radiation (PAR, $\mu mol~photons~m^{-2}~s^{-1}$).

Nutrient Uptake

Nutrient uptake rates ($VNH4$, $VNO3$, and $VPO4$) are cell-specific ($mmol~N~cell^{-1}~s^{-1}$ or $mmol~P~cell^{-1}~s^{-1}$) and include intracellular nutrient limitation (Droop limitation):

\[\begin{align} VNH4 &= VNH4_{cell}\cdot regQ_N\cdot\frac{[NH4]}{[NH4]+K_{NH4}^{sat}} \nonumber \\ VNO3 &= VNO3_{cell}\cdot regQ_N\cdot\frac{[NO3]}{[NO3]+K_{NO3}^{sat}} \nonumber \\ VPO4 &= VPO4_{cell}\cdot regQ_P\cdot\frac{[PO4]}{[PO4]+K_{PO4}^{sat}} \nonumber \end{align}\]

\[\begin{align} regQ_N &= \bigg[\frac{Nqmax-Q_N}{Nqmax - Nqmin}\bigg]_0^1 \nonumber \\ regQ_P &= \bigg[\frac{Pqmax-Q_P}{Pqmax - Pqmin}\bigg]_0^1 \nonumber \end{align}\]

\[\begin{align} Q_N &= (Nq + Bm \cdot R_{NC}) / (Cq + Bm) \nonumber \\ Q_P &= (Pq + Bm \cdot R_{PC}) / (Cq + Bm) \nonumber \end{align}\]

\[\begin{align} VNH4_{cell} &= VNH4max \cdot Sz^{VN_b} \cdot Bm \nonumber \\ VNO3_{cell} &= VNO3max \cdot Sz^{VN_b} \cdot Bm \nonumber \\ VPO4_{cell} &= VPO4max \cdot Sz^{VP_b} \cdot Bm \nonumber \end{align}\]

where $VNH4max$, $VNO3max$, and $VPO4max$ are constant parameters (see Parameters) while $VNH4_{cell}$, $VNO3_{cell}$, and $VPO4_{cell}$ are cell-specific maximum uptake rates ($mmol~N~cell^{-1}~s^{-1}$ or $mmol~P~cell^{-1}~s^{-1}$) depending on cell size, $Sz$.

Reserve Update

The model first updates C, N, and P reserves based on photosynthesis rate ($PS$) and nutrient uptake rates ($VNH4$, $VNO3$, and $VPO4$) as formulated above. The result is then used to calculate the biosynthesis and excretion rates.

\[\begin{align} Cq &= Cq+PS \cdot \Delta T \nonumber \\ Nq &= Nq+(VNO3+VNH4) \cdot \Delta T \nonumber \\ Pq &= Pq+VPO4 \cdot \Delta T \nonumber \end{align}\]

Biosynthesis And Excretion Rates

Potential biosynthesis rates are calculated based on C, N, P quotas as

\[\begin{align} BS_C &= Cq \cdot k_{mtb} \nonumber \\ BS_N &= Nq/R_{NC} \cdot k_{mtb} \nonumber \\ BS_P &= Pq/R_{PC} \cdot k_{mtb} \nonumber \end{align}\]

\[k_{mtb}= kmtb_{max} \cdot Sz^{kmtb_b}\]

The minimum of these rates gives the actual biosynthesis rate, $BS$ ($mmol~C~cell^{-1}~s^{-1}$), and the difference between carbon-based biosynthesis rate and $BS$ gives the excretion rate, $ExuC$ ($mmol~C~cell^{-1}~s^{-1}$).

\[BS = min(BS_C, BS_N, BS_P)\]

\[ExuC = BS_C - BS\]

Chlorophyll Synthesis

\[S_{chl} = \rho_{chl} * BS * R_{NC}\]

\[\begin{equation} \rho_{chl} = \begin{cases} chl:N * \frac{PC*Bm}{\alpha I \cdot chl}, & \quad \alpha I > 0,\\ 0 & \quad else. \end{cases} \nonumber \end{equation}\]

Respiration

\[Respir = respir_a \cdot Sz^{respir_b} \cdot Bm\]

Biomass Update

Biosynthesis yields a biomass increase, a commensurate reduction in nutrient reserves, and an increase in chlorophyll. The carbon reserve is further modified by respiration.

\[\begin{align} Bm &= Bm + BS \cdot \Delta T \nonumber \\ Cq &= Cq - (BS - Respir) \cdot \Delta T \nonumber \\ Nq &= Nq - BS*R_{NC} \cdot \Delta T \nonumber \\ Pq &= Pq - BS*R_{PC} \cdot \Delta T \nonumber \\ chl &= chl + S_{chl} \cdot \Delta T \nonumber \end{align}\]

Cell division

Relative cell size, $Sz$, is used to indicate cell division. Cells will not divide if $Sz<2.0$ and the probability of individual cell division ($P_D$) is then given by a sigmoidal function of $Sz$ with different cell division strategies.

\[Sz = (Bm + Cq) / Cquota\]

\[P_D = rand(Bernoulli(P_{dvid}*S_{dvid}))\]

where $S_{dvid}$ represents the sigmoidal function in different cell division strategies (listed below) and $P_D$ is computed every 10 min (when model time step is smaller than 10 min) or every time step (when model time step is larger than 10 min).

Sizer

The probability of cell division is a function of cell size and increases as the cell size gets larger.

\[S_{dvid} = tanh(stp_D * (Sz - reg_D)) + 1.0\]

Adder

The probability of cell division is a function of the difference between current cell size and initial cell size ($iSz$) and increases as the difference gets larger.

\[S_{dvid} = tanh(stp_D * (Sz - iSz - reg_D)) + 1.0\]

Timer

The probability of cell division is a function of clock time and increases in the dark.

\[S_{dvid} = tanh(stp_D * (t - reg_D)) + 1.0\]

Sizer+Timer

The probability of cell division is a multiplication of a function of clock time and a function of cell size.

\[S_{dvid} = (tanh(stp1_D * (Sz - reg1_D)) + 1.0) * (tanh(stp2_D * (t - reg2_D)) + 1.0)\]

Adder+Timer

The probability of cell division is a multiplication of a function of clock time and a function of the difference of current cell size and initial cell size.

\[S_{dvid} = (tanh(stp1_D * (Sz - iSz - reg1_D)) + 1.0) * (tanh(stp2_D * (t - reg2_D)) + 1.0)\]

Grazing

For now, the model does not have individuals of grazers. Instead, a quadratic grazing term is parameterized and implemented in the model to represent the grazing loss of cells. For each individual, the probability to be grazed ($P_G$) is calculated as below.

\[P_G = P_{graz} * pop\]

where pop is the number of individuals in the grid cell where the individual stays.

Parameters