When I look back on this project, I can say with confidence: I wasn’t assigned to solve a century-old puzzle. In fact, my “official” task was far more straightforward: build a database of all the photosynthesis–irradiance (PI) experiments conducted over the past 50 years, thanks to the efforts of many dedicated scientists at the Bedford Institute of Oceanography (Fisheries and Oceans Canada) in Dartmouth, Nova Scotia, Canada. It was meant to be a year-long exercise in patience, coding, and careful data analysis, contributing to one chapter of my PhD.
And yet, somewhere along the way, I found myself face to face with a fundamental inconsistency in the very models I was supposed to be cataloging. It bothered me enough that I couldn’t sleep on it and eventually had to put my math hat on!
A Century-old Challenge
For those outside the field, a quick step back: phytoplankton are the microscopic plants, the base of the marine food webs and a major engine of the carbon cycle. They capture sunlight and convert it into energy, driving nearly half of oceanic primary production.
To understand how phytoplankton respond to light, we study photosynthesis-irradiance (PI) curves using mathematical equations. These curves show how photosynthesis changes with increasing light. At low light, photosynthesis rises linearly. At moderate light, it levels off (plateau phase) and with further increase of lights it declines. This decline is called photoinhibition—think of it like a “sunburn” for phytoplankton.
When photoinhibition is absent, PI curves can be described reliably using light-saturating models (LSMs; Jassby & Platt, 1976). But when photoinhibition is present, the common practice is to multiply LSMs by an exponential decay function. This forces the curve to have a single maximum and prevents the model from capturing the plateau phase often observed in real data. Biologically, these equations assume that photoinhibition begins at very low light and progressively overwhelms photosynthetic capacity, conflating the photodamage and photoinhibition concepts!
This approach also creates another problem: it prevents scientists from directly estimating the maximum photosynthesis capacity from the data. Instead, it must be calculated indirectly from several parameters, making the result value less reliable. To avoid this, researchers often sidestep the problem entirely, cutting their data just before photoinhibition begins. The cost is that the decline itself gets ignored as it once thought to be “unimportant” or even impossible to quantify.
By analyzing ~4,000 open-ocean PI experiments (1973–2022), we found photoinhibition in ~50% of cases; of those, about 80% ± 5% displayed a plateau. Far from rare, this pattern is common.
Thinking Outside the Box
Biologically, we expect little photodamage at low light; as irradiance increases, damage accumulates faster than repair. Beyond a threshold, damage can outpace photosynthetic capacity—this is when photoinhibition sets in. In other words, photoinhibition shouldn’t appear until sufficiently high irradiance, after which photosynthesis begins to decline.
For decades, though, the standard way to include photoinhibition was to multiply light-saturating models (LSMs) by an exponential decay that decreases continuously with increasing light (following Steele 1962 and later formulations). That choice effectively forces a single peak and treats photoinhibition as if it begins almost immediately after light got available, violating biological expectation (Fig. 1, left panel).
After assembling a pool of published models and testing them against our database, I noticed an inconsistency: the classic exponential form often fit non-inhibited data better than the hyperbolic form, yet in the presence of photoinhibition, the extended hyperbolic performed better. That suggested we needed a different route. We explored modeling photoinhibition as a saturating function of the reciprocal of irradiance rather than an exponential decay with irradiance, thereby satisfying biological expectation (Fig. 2). With thoughtful feedback from my supervisor, Dr. Andrew Irwin, we reshaped the formulation so it can also estimate the onset of photoinhibition, the threshold irradiance at which photoinhibition rate reaches out to its maximum value, directly from the data. Statistical analysis strongly supports adopting our new parsimonious double-tanh photoinhibition model, which quantitatively and qualitatively outperforms existing models (Fig. 1, right panel).
Importance & Impact
Photosynthesis-irradiance (PI) models are core components of how we estimate ocean primary production. By capturing the plateau and photoinhibition simply and accurately, our formulation should lower uncertainty in primary-production estimates and improve ocean-climate predictions. Better estimates support more robust projections of the ocean’s role in Earth’s climate.
Building the foundation: database & software
Alongside the model, we assembled an open-access meta-dataset of PI experiments with photoinhibition metrics (spanning 1973–2022) and released piCurve, an open-source R package. The goal is simple: make the workflow standard, transparent, and easy to apply so others can use these methods on their own data.
piCurve provides automated steps to check data, classify curve types, fit models, and report parameters—letting users focus on the measurements rather than the equations. For the community, the dataset and toolkit can serve as a practical benchmark for PI-curve parameterization. We hope they help improve consistency, reveal where models disagree, and make results more reproducible across studies.
M. Amirian, M., Finkel, Z.V., Devred, E., Irwin, A.J. “Parameterization of photoinhibition for phytoplankton”. Commun Earth Environ 6, 707 (2025). https://doi.org/10.1038/s43247-025-02686-3
M. Amirian, M. and Irwin, A.J. “piCurve: an R package for modeling photosynthesis–irradiance curves”. arXiv preprint arXiv:2508.14321v1, (2025). https://arxiv.org/abs/2508.14321