Mathematical Modeling of Photo- and Thermomorphogenesis in Plants

Increased day lengths and warm conditions inversely affect plant growth by directly modulating nuclear phyB, ELF3, and COP1 levels. Quantitative measures of the hypocotyl length have been key to gaining a deeper understanding of this complex regulatory network, while similar quantitative data are the foundation for many studies in plant biology. Here, we explore the application of mathematical modeling, specifically ordinary differential equations (ODEs), to understand plant responses to these environmental cues. We provide a comprehensive guide to constructing, simulating, and fitting these models to data, using the law of mass action to study the evolution of molecular species. The fundamental principles of these models are introduced, highlighting their utility in deciphering complex plant physiological interactions and testing hypotheses. This brief introduction will not allow experimentalists without a mathematical background to run their own simulations overnight, but it will help them grasp modeling principles and communicate with more theory-inclined colleagues.

Generalized indicator-based determination of solution pH

pH indicators can be used both fast responsive as well as long-term stable sensors. They have been extensively used for monitoring pH changes in fast kinetic reactions as well as slowly changing pH in oceanic waters. If the pH range that needs to be covered is narrow it is possible to use only one indicator of appropriate protonation constant; otherwise, mixtures of two or more indicators are used for monitoring pH values covering a broad range of pH. In this paper we presented a new methodology for determining pH of solutions using mixtures of pH indicators. The pH calculation is based on the strict application of the basic laws of mass action and mass conservation. The proposed method was evaluated by the successful determination of the pH values of solutions containing three indicators (neutral red, phenol red (two different protonation constants), and methyl orange) covering a wide range of pH values from 0.5 to 9. The method was also applied for rapid monitoring of pH changes in stopped-flow measurements, investigating the reactions of CO₂ in aqueous amine solutions.

Activity-based analysis of potentiometric pH titrations

The discrepancy between concentrations and activities is a predicament well known to the analytical chemist. Because of the difficulty of determining activity coefficients, the standard technique for quantitative equilibrium studies is to work under a particular 'constant ionic strength' by adding an excess of an inert salt. Under such conditions, activity coefficients are approximately constant and can be taken into the equilibrium constants which are defined for the chosen ionic strength (I). Here we propose a fundamentally different approach. Throughout the numerical analysis of the titration data, activity coefficients for all individual species are approximated by well-known equations based on the work of Debye-Hückel. The computational analysis of the measurements strictly obeys the law of mass conservation and obeys the law of mass action only approximately. The main novelty is that now the addition of inert salts is no longer required and measurements are done at minimal I. Consequently, the thermodynamic equilibrium constants are now determined much more robustly based on experiments taken at low I. The approach has been tested and validated with the two very well investigated 3-protic phosphoric and citric acids. In summary: the technique of artificially keeping ionic strength constant has been replaced by improved computational analysis.

The different faces of mass action in virus assembly

The spontaneous encapsulation of genomic and non-genomic polyanions by coat proteins of simple icosahedral viruses is driven, in the first instance, by electrostatic interactions with polycationic RNA binding domains on these proteins. The efficiency with which the polyanions can be encapsulated in vitro, and presumably also in vivo, must in addition be governed by the loss of translational and mixing entropy associated with co-assembly, at least if this co-assembly constitutes a reversible process. These forms of entropy counteract the impact of attractive interactions between the constituents and hence they counteract complexation. By invoking mass action-type arguments and a simple model describing electrostatic interactions, we show how these forms of entropy might settle the competition between negatively charged polymers of different molecular weights for co-assembly with the coat proteins. In direct competition, mass action turns out to strongly work against the encapsulation of RNAs that are significantly shorter, which is typically the case for non-viral (host) RNAs. We also find that coat proteins favor forming virus particles over nonspecific binding to other proteins in the cytosol even if these are present in vast excess. Our results rationalize a number of recent in vitro co-assembly experiments showing that short polyanions are less effective at attracting virus coat proteins to form virus-like particles than long ones do, even if both are present at equal weight concentrations in the assembly mixture.

Modelling the growth of plants with a uniform growth logistics

BACKGROUND AND AIMS

The increment model has previously been used to describe the growth of plants in general. Here, we examine how the same logistics enables the development of different superstructures.

METHODS

Data from the literature are analyzed with the increment model. Increments are growth-invariant molecular clusters, treated as heuristic particles. This approach formulates the law of mass action for multi-component systems, describing the general properties of superstructures which are optimized via relaxation processes.

RESULTS

The daily growth patterns of hypocotyls can be reproduced implying predetermined growth invariant model parameters. In various species, the coordinated formation and death of fine roots are modeled successfully. Their biphasic annual growth follows distinct morphological programs but both use the same logistics. In tropical forests, distributions of the diameter in breast height of trees of different species adhere to the same pattern. Beyond structural fluctuations, competition and cooperation within and between the species may drive optimization.

CONCLUSION

All superstructures of plants examined so far could be reproduced with our approach. With genetically encoded growth-invariant model parameters (interaction with the environment included) perfect morphological development runs embedded in the uniform logistics of the increment model.