The Standard P Model

This page gives an overview of the calculations for the standard form of the P Model (Prentice et al., 2014, Wang et al., 2017) along with links to further details of the core components of the model. It may be useful to read this alongside:

The worked examples of using pyrealm to fit the Standard P Model.
The overview of the expected predictions of the model, given forcing data within normal bounds.
The descriptions of the key Python classes used to fit the standard P Model: PModelEnvironment and PModel.

Important

The standard form of the P Model is intended to work with data on greater than weekly timescales, where the plants are assumed to be roughly at equilibrium with their environment. At faster time scales, the subdaily form of the P Model accounts for more slowly responding systems that introduce acclimation lags into the model calculations.

The diagram below shows the workflow used for the calculation of the standard P Model: the input forcing data is show in blue, internal variables are shown in green and the core shared calculations are shown in red.

In overview:

The forcing variables are used to calculate the photosynthetic environment for the model, including the:
- photorespiratory compensation point (\(\Gamma^*\)),
- Michaelis-Menten coefficient for photosynthesis (\(K_{mm}\)),
- relative viscosity of water (\(\eta^*\)),
- the partial pressure of \(\ce{CO2}\) in ambient air (\(c_a\)), and
- the absorbed irradiance (\(I_{abs}\)).
The photosynthetic environment is then used to calculate optimal chi, given the method set by the method_optchi argument, to calculate:
- the ratio of internal to ambient \(\ce{CO2}\) partial pressure (\(\chi\)),
- the internal \(\ce{CO2}\) partial pressure (\(c_i\)),
- the \(\ce{CO2}\) limitation factors to both light assimilation (\(m_j\)) and carboxylation (\(m_c\)) along with their ratio (\(m_j / m_c\)).
The term \(m_j\) is at the heart of the P model and describes the trade off between carbon dioxide capture and water loss in photosynthesis. Given the environmental conditions, a leaf will adjust its stomata to a value of \(\chi\) that optimises this trade off. When \(\chi\) is less than one, the partial pressure of \(\ce{CO2}\) inside the leaf is lowered and \(m_j\) captures the resulting loss in light use efficiency.
The photosynthetic environment is also used to calculate the quantum yield of photosynthesis (\(\phi_0\)), given the method set by the method_kphio argument.
Theory suggests that \(m_j\) and \(m_c\) should be further limited (\(J_max\) limitation), with different approaches proposed (Smith et al., 2019, Wang et al., 2017). The calculation of limitation factors, given the method set by the method_jmaxlim argument, represents these alternative corrections as:
- a limitation term on the electron transfer rate (\(f_j\)), and
- a similar limitation term on the carboxylation capacity (\(f_v\)).
From these values, and the molar mass of carbon (\(M_C\)), the light use efficiency of photosynthesis can then be calculated as:

\[ \text{LUE} = M_C \cdot \phi_0 \cdot m_j \cdot f_v \]

The gross primary productivity (GPP) is then simply the product of LUE and the absorbed irradiation (\(I_abs = f_{APAR} \cdot \text{PPFD}\)).
The approach above is equivalent to directly calculating the minimum of the assimilation rates given limitation of the electron transfer pathway (\(A_j\)) and limitation of the carboxylation pathway (\(A_c\)). However the model does still estimate the maximum rates of electron transfer (\(J_{max}\)) and carboxylation capacity (\(V_{cmax}\)) and then uses Arrhenius scaling to estimate those values at standard temperature (\(J_{max25}\)) and carboxylation capacity (\(V_{cmax25}\)).