Developmental polymorphism and the Brooks-Dyar law
Authors: Rivera Fernando , Vasquez Diego , Ruiz Carolina , Gómez Valderrama Juliana , Borrero Echeverry Felipe , Rincón Diego 
Published: June 2023

Many protocols for research in insect biology and ecology require accurate determination of larval instar. The Brooks-Dyar Law (1886, 1890) states that the measurement of sclerotized structures follows a predictable regular geometric progression that can be used to determine accurately both larval instar of single larvae and the number of instars before pupation in a population. The Brooks-Dyar Law has been used extensively in studies of a number of holometabolous and hemimetabolous orders. Although the Brooks-Dyar’s Law describes the variation in size among insect larvae as a function of development, the mathematical formula of the law has only been defined empirically, without any insights on the biological meaning of parameters (but see Hutchinson et al. [1997]). Moreover, the current definition assumes that insects go through a fixed number of instars before pupation, which is not always the case for many insect orders (Esperk et al. 2007). Developmental polymorphism describes how environmental factors, such as food quality, can alter the development of certain species and vary the number of larval instars. Since developmental polymorphism seems to be common in insects (Etilé and Despland 2008), the Brooks-Dyar Law needs to be adjusted to account for epigenetic factors, such as food quality, which may affect the number of instars before pupation. Furthermore, a mechanistic mathematical definition of the Brook-Dyar’s law would be useful to more accurately describe the relative properties of a given epigenetic factor. We used the fall armyworm (Spodoptera frugiperda) as a model to test if the Brooks-Dyar Law holds true for larvae reared on different natural and artificial diets. We studied whether the distance between frontal setae can be used as a reliable measurement for the application of the Brooks-Dyar Law. Finally, we propose a mathematical formula for the Brooks-Dyar’s law, which includes a constant and two parameters, all defined in light of food quality as a driving epigenetic factor Many protocols for research in insect biology and ecology require accurate determination of larval instar. The Brooks-Dyar Law (1886, 1890) states that the measurement of sclerotized structures follows a predictable regular geometric progression that can be used to determine accurately both larval instar of single larvae and the number of instars before pupation in a population. The Brooks-Dyar Law has been used extensively in studies of a number of holometabolous and hemimetabolous orders. Although the Brooks-Dyar’s Law describes the variation in size among insect larvae as a function of development, the mathematical formula of the law has only been defined empirically, without any insights on the biological meaning of parameters (but see Hutchinson et al. [1997]). Moreover, the current definition assumes that insects go through a fixed number of instars before pupation, which is not always the case for many insect orders (Esperk et al. 2007). Developmental polymorphism describes how environmental factors, such as food quality, can alter the development of certain species and vary the number of larval instars. Since developmental polymorphism seems to be common in insects (Etilé and Despland 2008), the Brooks-Dyar Law needs to be adjusted to account for epigenetic factors, such as food quality, which may affect the number of instars before pupation. Furthermore, a mechanistic mathematical definition of the Brook-Dyar’s law would be useful to more accurately describe the relative properties of a given epigenetic factor. We used the fall armyworm (Spodoptera frugiperda) as a model to test if the Brooks-Dyar Law holds true for larvae reared on different natural and artificial diets. We studied whether the distance between frontal setae can be used as a reliable measurement for the application of the Brooks-Dyar Law. Finally, we propose a mathematical formula for the Brooks-Dyar’s law, which includes a constant and two parameters, all defined in light of food quality as a driving epigenetic factor