Plant responses to plant growth regulators (PGRs) often include increases in size, biomass, and yield, thereby differentiating them from other pesticides (e.g., herbicides, fungicides, insecticides), which often include damage, inhibition, or lethality to their targeted organisms (Pohanish, 2014). Such responses may be amplified or diminished when PGR compounds are applied as mixtures. Existing methods and formulas based on proportions or percent damage, inhibition, or lethality cannot be directly and readily applied to PGR mixtures for the aforementioned “increase” responses; therefore, a different approach is required.
The methods used to quantify the predicted effect of a mixture on biological systems have been studied in great depth and are considered paramount for accurately assessing synergy. Various methods of analyzing the effects of mixtures have been used, debated,and repeatedly reviewed over the past century (Cedergreen et al., 2014; Foucquier and Guedj, 2015). Two distinct, well-known, and widely accepted theories for predicting mixture effects were proposed by Loewe and Muischnek (1926) and Bliss (1939). Loewe and Muischnek’s theory assumed that all active components in a mixture act on the same biological target but with different potencies to achieve an equivalent result; therefore, one component could be substituted at a constant proportion to the other. This concept has been given many titles, such as Loewe Additivity, Dose Addition, the Additive Dose Model (ADM), Concentration Addition (CA), or Similar Joint Action, depending on the authors and their research areas (Cedergreen et al., 2013). Both isobolograms and combination indices exemplify approaches that were developed based on Loewe and Muischnek’s theory (Gisi, 1996; Kull et al., 1961; Roell et al., 2017; Tammes, 1964; Voorspuij and Nass, 1957). Loewe and Muischnek’s theory is a topic that continues to be actively studied and reviewed by researchers in many fields, including pharmacology (Chou, 2006; Foucquier and Guedj, 2015; Roell et al., 2017; Zhao et al., 2010).
Alternatively, Bliss (1939) proposed the theory of Independent Joint Action using insects as a model organism for which mortality was the response to the components of a mixture. The Independent Joint Action theory assumes the active components in a mixture function independently with different modes of action. Therefore, the component effects of a mixture can be predicted by the dosage-mortality of each component when applied alone. Bliss’s formula to predict the effect of a two-component mixture is presented as follows:where P(A+B) signifies the predicted effect as the proportion killed as a result of the mixture of components A and B at rates a and b, respectively; PA and PB indicate the proportion killed as a result of component A at rate a and component B at rate b when each is applied alone (Bliss, 1939). Additionally, (1 − PA) PB can be interpreted as the action of component B when acting on what survived the action of component A. As Bliss (1939) pointed out, the aforementioned formula, after transformation, is tantamount to Abbott’s formula (Abbott, 1925), which was first introduced to compute the percent control of insects as an adjustment to exclude natural insect mortality during experiments.
Similar to Loewe and Muscinek’s theory, the Independent Joint Action theory has been assigned many names, including Bliss Independence, Independent Action, Multiplicative Survival Model (MSM), Response Multiplication, Response Addition, and Effect Addition (Bliss, 1939; Cedergreen et al., 2014; Colby, 1967; Gisi, 1989, 1996; Levy et al., 1986; Morse, 1978; Nash, 1981; Zhao, et al., 2014). Bliss’s formula is widely applicable and frequently used to assess component interactions by determining the predicted effect of mixtures tested during agricultural experiments (e.g., insecticides, herbicides, and fungicides) and environmental research studies (e.g., toxicity assessments) in which effects are expressed as proportions or in terms of percent damage, inhibition, or lethality (Altenburger, et al., 2013; Foucquier and Guedj, 2015; U.S. Environmental Protection Agency,2000). Bliss’s Independent Joint Action theory is also regarded as a particularly important alternative to Loewe Additivity in pharmacological research (Zhao et al., 2014). Despite its wide acceptance and frequent use, Bliss’s formula has limitations. It is limited to proportional data types that range from 0 to 1 (or percentage data ranging from 0% to 100%). Additionally, Bliss’s formula cannot accommodate nonproportional data, such as growth or increases in size or weight (Foucquier and Guedj, 2015). Because Bliss’s formula is not designed for computing predicted nondestructive effects, many responses to plant growth regulators and stimulants cannot be used in Bliss’s formula to assess synergy. Therefore, a formula other than Bliss’s is needed for computing predicted effects on growth for mixtures of plant growth regulators or combinations of growth-affecting components/factors such as fertilizers, temperature, moisture, and plant growth-promoting microbes (PGPM).
A formula derived from Bliss’s Independent Joint Action theory is presented here. A single data set from various PGR mixture studies was selected and used solely for demonstrating the application of the proposed formula. During the selected experiment, a mixture of plant growth regulators (S-abscisic acid and gibberellic acid) was tested. Although S-abscisic acid (S-ABA) can have an inhibitorygrowth effect, gibberellins are known to stimulate cell elongation (Basra, 2000; Fletcher et al., 2000; Kaur et al., 2018). Accordingly, S-ABA and gibberellic acids arguably belong to different classes of plant growth regulators.
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