Dušan Lazár’s lecture on mathematical models of photosynthesis
David Fuente’s lecture on the BDM2 model to simulate dynamics in oscillating light
Wolfram Mathematica BDM2 core model v3
MATLAB BDM2 core model v3
Python BDM2 core model v3
The system’s independent variables in BDM2 were:
- Fraction of the oxidized plastoquinone (PQ)
- Fraction of the oxidized donors of PSI (PIox)
- The proton concentration in the lumen (HL)
- ATP
- Zeaxanthin
- Activated PsbS
Dependent variables were:
- Chlorophyll fluorescence
- Oxygen evolution rate
- NPQ
Mathematical Models of Photosynthesis Under Dynamic Light
Modeling photosynthesis under dynamic light conditions is less common than modeling under dark-to-constant light transitions. For reviews on the latter, see e.g., Lazár and Schansker (2009) and Stirbet et al. (2020). Nonetheless, many models have been developed to simulate photosynthesis under dynamic light. Here, “dynamic light” refers to any temporally varying illumination — including terms used in the literature such as flashing light, pulsed light, intermittent light, light-dark cycles, fluctuating light, and, in our case, harmonically oscillating light.
A summary of these models is presented in Lazár’s lecture, and a full review on this topic is currently in preparation by the author.
Spontaneous Damped Oscillations
Focusing specifically on oscillatory phenomena, many mathematical models have been developed to describe spontaneous damped oscillations. These oscillations are typically triggered by perturbing a steady state — either by a sudden increase in light intensity or more commonly by an abrupt rise in CO₂ concentration, e.g., to 2% (Lazár et al., 2005).
Several hypotheses have been proposed to explain the source of these oscillations — sometimes referred to as an “oscillophore” (Lazár et al., 2005):
- ATP insufficiency due to competition between 3-phosphoglycerate kinase (PGK) and ribulose-5-phosphate kinase (RPK) during the Calvin cycle (Walker et al., 1983).
- Orthophosphate (Pi) availability affects oscillation amplitude, suggesting a significant regulatory role (Walker and Sivak, 1985; Sivak and Walker, 1986).
- Feedback inhibition of cytosolic fructose 1,6-bisphosphate phosphatase (FBPase) by fructose 2,6-bisphosphate, with delays introduced by other reactions in cytosolic sucrose synthesis (Stitt et al., 1988).
- Imbalance in ATP and NADPH supply to the Calvin cycle, caused by inefficient photosynthetic electron transport (Laisk et al., 1991).
- Rubisco involvement (ribulose-1,5-bisphosphate carboxylase/oxygenase) as the source of oscillations, supported by both theoretical analysis (Ryde-Pettersson, 1991; Roussel, 1998) and experimental evidence — e.g., transgenic tobacco plants with reduced Rubisco showed no oscillations in fluorescence or O₂ evolution (Stitt et al., 1991). Further support on the role of Rubisco and its activation in the spontaneous oscillations came from measurement of the spontaneous oscillations under different temperatures combined with theoretical approaches (Lazár et al., 2005).
These hypotheses have been explored via various mathematical models with differing success (see Giersch, 1986; Hahn, 1986; Laisk and Walker, 1986, 1989; Horton and Nicholson, 1987; Giersch and Sivak, 1991; Kukushkin, 1997; Roussel, 1998); Lazár et al., 2005, among others).
The collective conclusion is that there is likely no single “oscillophore” — instead, multiple interconnected processes contribute to oscillatory behavior. This is supported by Fourier transform analysis of fluorescence oscillations (Ferimazova et al., 2002) and aligns with insights from Metabolic Control Analysis (MCA).
MCA is a mathematical framework that quantifies the influence of system components on measurable variables. Although initially formulated for steady-state systems, MCA has since been adapted to oscillating and periodically forced systems. Its summation theorems imply that all system components influence any given variable — the key question is to what extent each contributes.
Thus, oscillations in photosynthesis are not caused by a single reaction but arise from the dynamics distributed of the entire system. This belongs to the attributes of emerging phenomena. An example of applying MCA to a spontaneous oscillation model is found in Lazár et al., 2005.
Forced Oscillations
The earlier, highly reduced phenomenological models of forced oscillations aimed at identifying mechanisms that can simulate the appearance of the upper harmonic modes in responses to forcing:
- Nedbal and Březina (2002) suggested that the upper harmonic modes appear due to the negative feedback regulation by NPQ.
- Nedbal et al. (2005) simulated potential appearance of the upper harmonic modes due to delays in biochemical processes.
- Nedbal and Lazár (2021) used a reduced model to find an analytical solution that exhibits the upper harmonic modes.
The recent mechanistic mathematical models could capture multiple features of photosynthetic responses to oscillating light, including the upper harmonics.
- The Basic DREAM Model (BDM) introduced in Fuente et al. (2024). BDM includes regulation by non-photochemical quenching (NPQ) via a single, unspecified quencher.
- Its refinement, BDM2, developed in Niu et al. (2025). BDM2 improves on this by modeling two identified quenchers: protonated PsbS protein and zeaxanthin.
Both models were used to compare simulations with experimental data collected under low- and high-amplitude oscillating light, respectively.
BDM2 is discussed in detail in the lecture by David Fuente.