Predicting plant growth response under fluctuating temperature by carbon balance modelling

The study addresses how to quantitatively describe and predict dynamic metabolic responses of plants to transient environmental heat stress. Traditional kinetic ordinary differential equation (ODE) models require extensive enzyme parameters and may be difficult to apply under rapidly fluctuating conditions. Prior spline-based approaches fit time series accurately but lack direct biological interpretability and predictive power. The authors propose a Fourier polynomial-based framework to capture oscillatory dynamics in photosynthesis and carbohydrate metabolism, aiming to integrate net photosynthetic CO₂ assimilation with starch and sugar turnover and to predict impacts on carbon balance and growth during transient heat exposure.

Kinetic ODE/PDE models have been widely used to simulate metabolic processes (e.g., in microbial engineering, disease, and plant metabolism), but are limited by parameter availability and structural uncertainties. Resources like KiMoSys aggregate metabolite, protein, and flux data to aid modeling. Flux analysis and tracer studies estimate pathway activities without resolving individual enzyme kinetics and can be integrated with metabolite levels. Previous work used spline interpolations to derive metabolic functions from time-series data to identify regulatory cascades, though these are less predictive. Fourier analysis has been applied to biological time series (e.g., gene expression) and can be coupled with machine learning. Plant heat responses often involve reduced photosynthesis due to Rubisco deactivation rather than PSII damage; acclimation mechanisms and roles of hexose phosphorylation enzymes (hexokinase, fructokinase) have been implicated in stress responses.

- Modeling framework: Developed a block diagram model of central carbohydrate metabolism using Fourier polynomials fitted to experimental time-series data. Three input functions describe net photosynthesis (NPS), starch amount, and soluble sugar amount as Fourier polynomials (Eqs. 1–3). Balance equations compute net carbon availability: BE₁ = (1/6) × (NPS − d/dt(starch)), and BE₂ = BE₁ − d/dt(sugars) (Eqs. 4–5), with rates in µmol C6 h⁻¹ gDW⁻¹. Differentiation of starch and sugar Fourier polynomials yields biosynthesis rates; integration provides accumulated carbon balances. - Biological material: Arabidopsis thaliana accession Columbia-0 (Col-0), starch-deficient mutant pgm1 (AT5G51820), and sucrose phosphate synthase mutant spsa1 (AT5G20280; activity reduced to 30–50% of Col-0). - Growth and treatments: Plants grown under short-day 8h/16h (100 µmol m⁻² s⁻¹; 22/18 °C; 60% RH). On sampling day, temperature set to 22 °C for 0–2 h light, then increased for 4 h to 32, 36, or 40 °C (2–6 h), followed by recovery at 22 °C (6–8 h). Control remained at 22 °C throughout. Samples collected at 0, 2, 6, and 8 h. - Gas exchange and chlorophyll fluorescence: Net CO₂ assimilation and transpiration measured with WALZ GFS-3000FL (head 3010-S) under cabinet-matched conditions. PSII maximum quantum yield (Fv/Fm) and rapid light curves (ETR, qP, qN) measured with WALZ Junior-PAM after 15 min dark adaptation; saturating pulses and actinic steps up to 2250 µmol photons m⁻² s⁻¹. - Metabolite quantification: Starch and soluble sugars (sucrose, glucose, fructose) extracted from lyophilized leaf tissue. Starch hydrolyzed (NaOH, amyloglucosidase) and quantified via glucose oxidase/peroxidase assay. Sucrose quantified by anthrone assay after KOH treatment; glucose by hexokinase/G6PDH (NADPH at 340 nm); fructose quantified after adding phosphoglucoisomerase. SPS activity measured via anthrone after incubation with substrates (UDP-glucose, F6P, G6P). - Data analysis: Fourier series fitting in MATLAB; block diagrams in Simulink. Time derivatives and integrals computed analytically/numerically from Fourier polynomials. Statistics in R/RStudio. Leaf rosette surface quantified with Fiji (SIOX plugin). - Growth validation: Separate experiment measured relative increase in leaf surface over 7 days: control (7 d at 22/18 °C) vs heat regime (3 d at 40/24 °C followed by 4 d at 22/18 °C). n ≥ 10 per genotype. - Replication and design: n=3 for gas exchange per genotype/condition; n ≥ 3–6 for fluorescence; n ≥ 3–5 for metabolite assays depending on genotype; randomized pot positions; blinded data collection.

- Fourier polynomial fits captured net CO₂ assimilation dynamics with high accuracy (R² > 0.94 for most cases; pgm1 at 32 °C: R² = 0.8177). - Heat effects on net photosynthesis: Temperature increases to 32–40 °C caused an initial drop in CO₂ assimilation that later stabilized. Col-0 showed significantly decreased NPS in the latter half of the 36 and 40 °C treatments, while spsa1 was comparatively robust; pgm1 was most susceptible during heat and recovery transitions. - Transpiration increased with temperature, peaking at 40 °C at roughly threefold the 22 °C rates, and decreased upon recovery; no genotype differences detected. - Photosystem performance: Fv/Fm in Col-0 decreased only during recovery from 40 °C; pgm1 decreased during 40 °C then recovered; spsa1 showed no significant Fv/Fm change. At 40 °C, spsa1 had significantly higher ETR and qP across PPFD, indicating altered photosystem/thylakoid behavior; Col-0 showed increased qP; pgm1 was least affected, with a qP drop upon recovery. - Starch dynamics: Heat caused significant starch decreases at 6 h in Col-0 and spsa1 (P < 0.001). During recovery (6–8 h), starch increased by ~40% after 32 °C and by >90% after 36 °C; Col-0 at 36 °C reached control levels by 8 h. pgm1 starch was below detection. - Sucrose dynamics: No significant changes at 32 °C. At 40 °C, Col-0 sucrose increased at 6 h (P < 0.001). pgm1 accumulated more sucrose overall but heat (36–40 °C) reduced sucrose at 6 h (P ≈ 0.05), recovering by 8 h. spsa1 had high variance at lower temperatures; at 40 °C, sucrose was significantly higher than Col-0 at 6 and 8 h (P < 0.001) and higher than spsa1 control (6 h: P < 0.004; 8 h: P < 0.01). - Hexoses: pgm1 showed markedly elevated glucose/fructose over the day compared to Col-0 and spsa1. Col-0 and spsa1 glucose dropped at 32–36 °C (P < 0.001); recovery restored (32 °C) or elevated (36–40 °C) glucose vs control. Fructose trends were more conserved; in spsa1, significant increases appeared only at 8 h for 36–40 °C (P < 0.001). In pgm1, hexoses decreased at 6 h under heat (lowest at 40 °C; P < 0.001) with partial recovery by 8 h for 36–40 °C. - Balance equation dynamics: Time derivatives of BE₁/BE₂ showed damped oscillations with increasing temperature; pgm1 exhibited larger fluctuation amplitudes during recovery than Col-0; spsa1 displayed notable damping at 36 °C. Fundamental frequencies of BE polynomials showed genotype- and temperature-specific patterns (e.g., Col-0 frequencies doubled at 32–36 °C then dropped at 40 °C; pgm1 BE₁ frequencies increased with temperature; spsa1 lowest at 36 °C). - Integrals (carbon gain): Heat reduced integrated NPS in Col-0 and pgm1, with strongest deviations mid-treatment; spsa1 assimilation under heat closely matched control. Including starch (BE₁) minimized discrepancies in Col-0 (32 and 40 °C similar to control), not observed in pgm1; spsa1 BE₁ integrals increased under heat. Including sugars (BE₂) further minimized discrepancies early in Col-0; pgm1 showed reduced net carbon flux into sugars under heat, compensating reduced assimilation; spsa1 increases were modest. - Growth validation: Over 7 days, transient heat (3 d at 40/24 °C + 4 d recovery) significantly reduced relative leaf surface increase in Col-0 and pgm1 (stronger in pgm1), but not in spsa1, consistent with BE integrals and model predictions.

The Fourier polynomial-based balance model integrated NPS with starch and sugar turnover to quantify carbon balance dynamics under transient heat. The differential damping of derivative oscillations and shifts in fundamental frequencies across genotypes highlight distinct metabolic buffering capacities. Starch biosynthesis appears to stabilize NPS under heat, as starchless pgm1 showed destabilized assimilation and greater fluctuation amplitudes, while spsa1 exhibited robust NPS and carbon balance despite reduced sucrose biosynthesis capacity—potentially due to enhanced starch accumulation or altered photosystem organization. Similar transpiration across genotypes argues against stomatal differences as the primary driver. Conserved fructose dynamics suggest prominent glycolytic consumption and hexose phosphorylation under heat. Importantly, integrals of BE₁/BE₂ predicted whole-plant performance, with spsa1 showing minimal heat-induced changes in carbon gain and no significant growth reduction, in contrast to Col-0 and pgm1. This demonstrates that Fourier polynomial modeling can capture short-term metabolic dynamics that translate into longer-term growth outcomes under fluctuating environments.

This work introduces a Fourier polynomial-based carbon balance modeling framework that accurately captures plant metabolic dynamics under transient heat and links them to growth outcomes. By combining NPS with starch and sugar dynamics in balance equations, the model revealed that reduced sucrose biosynthesis capacity and enhanced starch biosynthesis can stabilize carbon assimilation under heat. Genotype-specific signatures in derivatives, frequencies, and integrals enabled discrimination of metabolic responses and successfully predicted growth impacts. Future research could dissect the mechanistic bases of NPS stabilization (e.g., roles of Rubisco/activase, photorespiration, respiration), expand to additional genotypes and stress regimes, integrate with machine learning for large-scale datasets, and explore broader applications of Fourier-based approaches in nonlinear metabolic modeling.

- Mechanistic inferences (e.g., effects on Rubisco or Rubisco activase, respiration) are speculative and not directly tested in this study. - One genotype (pgm1) showed a lower fit quality for NPS at 32 °C (R² = 0.8177), indicating modeling limits for some conditions. - Sucrose measurements in spsa1 exhibited high variance except at 40 °C, potentially affecting statistical power at lower temperatures. - Growth validation used a modified heat regime (3 days at 40/24 °C) differing from the 4 h transient treatment, which may limit direct comparability despite qualitative agreement. - The approach relies on time-series experiments in a single species (Arabidopsis thaliana) and specific mutants; generalizability to other species or environmental contexts remains to be established.