Mathematical modeling is crucial for quantifying system dynamics in biology. Ordinary differential equations (ODEs) are often used to model metabolic processes, but require detailed kinetic parameters that are often difficult to obtain. Alternative approaches, such as flux analysis, provide insights into metabolic states but offer limited information on individual enzyme activities. Previous research by the authors utilized spline interpolation to estimate metabolic functions, but these functions lack biological relevance and predictive power. This study proposes a new approach using Fourier polynomials to model the dynamic interactions between plant metabolism and fluctuating environmental conditions, specifically focusing on the effects of transient heat stress on photosynthesis and carbohydrate metabolism in Arabidopsis thaliana. The rationale is that Fourier polynomials can accurately capture oscillatory processes, a common feature in biological systems. The ability to accurately model and predict plant responses to dynamic environmental changes is crucial for optimizing agricultural practices and understanding plant adaptation in a changing climate.

The paper reviews existing methods for modeling metabolic dynamics, including ODE-based kinetic models and flux analysis. It highlights the limitations of ODE models, such as the need for extensive and often unavailable kinetic parameters, and the limitations of flux analysis in providing detailed information about individual enzyme activities. The authors mention their prior work using spline interpolation for analyzing time-series data, but acknowledge its limitations in terms of biological interpretability and predictive power. This lays the groundwork for introducing their novel Fourier polynomial-based approach, highlighting the advantages of this method in capturing oscillatory patterns and providing a more biologically relevant and predictive model for plant metabolism under dynamic conditions.

The researchers developed a block diagram model of central carbohydrate metabolism in Arabidopsis thaliana, incorporating experimental data on net photosynthesis (NPS), starch, and sugar metabolism. The model utilized Fourier polynomials to represent the dynamic changes in these variables over time. Three input functions (FPinput) were defined for NPS, starch, and sugars, each represented by a Fourier polynomial (Equations 1-3). Two balance equations (BE1 and BE2) were derived to describe carbon balance, integrating NPS rates and the rates of starch and sugar synthesis and degradation (Equations 4-5). The parameters of the Fourier polynomials were estimated from experimental data. The study used three Arabidopsis genotypes: wild-type (Col-0), a starch-deficient mutant (*pgm1*), and a sucrose biosynthesis-deficient mutant (*spsa1*). Plants were subjected to transient heat stress (32°C, 36°C, 40°C) for 4h, with measurements taken at 0h, 2h, 6h, and 8h. Net CO2 assimilation rates were measured using a gas exchange system. Chlorophyll fluorescence parameters (Fv/Fm, ETR, qP, qN) were measured using pulse-amplitude modulation (PAM) fluorometry. Starch, sucrose, glucose, and fructose concentrations were quantified using photometric assays. The activity of sucrose phosphate synthase (SPS) was also determined. Finally, a long-term growth experiment (7 days) with 3-day heat stress was conducted to assess the impact of the transient heat treatment on overall plant growth, using leaf rosette surface area as a proxy. Statistical analysis was performed in R and MATLAB. Fourier series fitting, block diagram modelling in Simulink, and image analysis were also employed. The authors provide detailed descriptions of all experimental protocols and analytical methods used.

The Fourier polynomial model accurately captured the dynamics of net CO2 assimilation, starch, and sugar metabolism in all three Arabidopsis genotypes under different temperature regimes (R² > 0.94 in most cases). Transient heat stress significantly affected net CO2 assimilation rates, with the starch-deficient *pgm1* mutant being the most susceptible. The *spsa1* mutant exhibited relatively stable photosynthesis under heat stress. Analysis of chlorophyll fluorescence parameters indicated that the effects of heat stress on net CO2 assimilation were not primarily due to photosystem damage, but rather to other factors such as rubisco deactivation. Transient heat stress also significantly altered starch and soluble carbohydrate dynamics, with varying responses among the genotypes. The derivatives of the balance equations revealed genotype-dependent system fluctuations in response to heat stress, with oscillations being dampened in Col-0 and *pgm1* under higher temperatures. Integrals of the balance equations, representing the net carbon gain over time, showed that starch dynamics in Col-0 adjusted proportionally to affected NPS rates, mitigating the impact of heat stress. In contrast, the starch-deficient *pgm1* showed a more pronounced reduction in net carbon flux. The *spsa1* mutant had the most robust carbon assimilation and carbohydrate metabolism under heat stress. Finally, the long-term growth experiment showed that the leaf rosette surface area of Col-0 and *pgm1* were significantly reduced by transient heat stress, while *spsa1* exhibited no significant difference.

The study demonstrates that the Fourier polynomial-based model effectively captures the dynamic interactions between plant metabolism and environmental conditions, specifically transient heat stress. The results support the hypothesis that reduced sucrose biosynthesis capacity and increased starch biosynthesis capacity can stabilize carbon assimilation under heat stress. The different responses of the three genotypes highlight the importance of considering genotype-specific responses to environmental stress. The use of Fourier analysis not only provides a quantitative description of metabolic dynamics but also reveals important insights into system properties such as stability and oscillation patterns. The model's ability to predict plant growth performance based on short-term metabolic responses suggests its potential value in agricultural applications and in predicting plant responses to climate change.

This research introduces a novel approach using Fourier polynomials to model plant carbon metabolism under dynamic temperature fluctuations. The model accurately predicted the effects of transient heat stress on photosynthesis and carbohydrate metabolism in Arabidopsis thaliana, revealing genotype-specific responses. The study highlights the importance of starch and sucrose metabolism in buffering against heat stress and underscores the potential of Fourier analysis for modelling complex biological systems. Future studies could explore the applicability of this approach to other plant species and environmental stresses, investigate the underlying molecular mechanisms responsible for the genotype-specific responses, and integrate this approach with machine learning algorithms for large-scale data analysis.

The study focuses on a limited number of Arabidopsis genotypes and temperature treatments. The model's accuracy might vary with different environmental conditions or plant species. The leaf surface area used as a proxy for plant growth may not capture all aspects of plant performance. Future research could address these limitations by expanding the scope of the study to include a wider range of genotypes, environmental conditions, and growth parameters.