Plants, algae, and fungi are essential for nearly all life on earth. Through photosynthesis, plants and algae convert solar energy to chemical energy in the form of organic compounds that sustains essentially all life on earth. In addition, plants and algae convert the carbon dioxide produced by respiring organisms to oxygen that is needed for respiration. Fungi decompose complex organic compounds produced by respiring organisms so that molecules can be recycled in photosynthesis and

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... . Plants, algae, and fungi have one important feature in common, their cells have walls. Expansive growth and its regulation are central to the life and development of plant, algal, and fungal cells, i.e. cells with walls. In recent decades there has been an explosion of information relevant to expansive growth of cells with walls. Mathematical models have been constructed in an attempt to organize and evaluate this information, to gain insight, to evaluate hypotheses, and to assist in the selection and development of new experimental studies. In this article some of the mathematical models constructed to study expansive growth of cells with walls are reviewed. It is nearly impossible to review all relevant research conducted in this area. Instead, the review focuses on the development of mathematical equations that have been used to model expansive growth, morphogenesis, and growth rate regulation of cells with walls. Also, relevant experimental findings are reviewed, conceptual models are presented, and suggestions for future research are proposed. The authors have attempted to provide an overview that is accessible to researchers that are not working in this field. Growth models for plant and fungal cells that describe expansive growth, morphogenesis, and growth rate regulation of plant, algal and fungal cells, e.g. [14-16, 24, 27-29, 35, 38, 45, 46, 63, 73]. Importantly, expansive growth of plant, algal, and fungal cells (cells with walls) employ the same physical principles. The cell transports or produces active solutes inside the plasma membrane and absorbs water from its surroundings through the process of osmosis. The absorption of water produces turgor pressure that stresses the wall. The wall is biochemically loosened, reducing both wall stresses (stress relaxation) and turgor pressure (turgor pressure relaxation). More water is absorbed in response to the decrease in turgor pressure, extending (deforming) the loosened wall. The wall deformation produces an increase in wall surface area (expansion) and a thinner wall. New wall polymers and other wall materials are added to maintain a nearly constant wall thickness. The series of events (i.e. wall loosening, wall stress relaxation, turgor pressure relaxation, water uptake, wall expansion, an increase in turgor pressure, an increase in wall stresses, and wall loosening again) is repeated continually during expansive growth. The wall deformation behavior during expansive growth depends on the wall's biochemically-mediated mechanical properties, but both irreversible and reversible (elastic) wall deformations occur simultaneously. Mathematically, a constitutive equation relates wall deformation (strain) behavior to wall stresses produced by the turgor pressure. The pressure probe is an instrument that can directly measure and control the magnitude of the turgor pressure in single plant, algal, and fungal cells. Pressure probe experiments demonstrate that controlling the magnitude of the turgor pressure and/or the wall mechanical properties can regulate expansive growth rate, e.