Fourier transform infrared (FT-IR) spectroscopy was used to monitor changes in the secondary structure of wheat prolamins, the main components of gluten, during mechanical deformation in a series of cycles of extension and relaxation. A sample derived from protein bodies isolated from developing grain showed a buildup of persistent beta-sheet structure. In gluten, the ratio of beta-sheet to random and beta-turn structures changed on extension. After the applied force was released, the sample recovered some of its original shape and structure, but the material became stiffer in consecutive extension cycles. The relationship between gluten structure and mechanical properties is discussed in terms of a model in which conversion of beta-turn to beta-sheet structure is a response to extension and a means by which elastic energy is stored in the system.

We examine critically the claims made by Fredrickson and Losada (2005) concerning the construct known as the "positivity ratio." We find no theoretical or empirical justification for the use of differential equations drawn from fluid dynamics, a subfield of physics, to describe changes in human emotions over time; furthermore, we demonstrate that the purported application of these equations contains numerous fundamental conceptual and mathematical errors. The lack of relevance of these equations and their incorrect application lead us to conclude that Fredrickson and Losada's claim to have demonstrated the existence of a critical minimum positivity ratio of 2.9013 is entirely unfounded. More generally, we urge future researchers to exercise caution in the use of advanced mathematical tools such as nonlinear dynamics and in particular to verify that the elementary conditions for their valid application have been met. Keywords:Positivity ratio, broaden-and-build theory, positive psychology, nonlinear dynamics, Lorenz system. Running head: THE CRITICAL POSITIVITY RATIO 3The "broaden-and-build" theory (Fredrickson, 1998(Fredrickson, , 2001(Fredrickson, , 2004 postulates that positive emotions help to develop broad repertoires of thought and action, which in turn build resilience to buffer against future emotional setbacks. Fredrickson and Losada (2005) took the broaden-and-build theory a step further, by proposing that an individual's degree of flourishing could be predicted by that person's ratio of positive to negative emotions over time, which they termed the "positivity ratio" (p. 678).On its own, the positivity ratio as propounded by Fredrickson and Losada (2005) is not a particularly controversial construct; indeed, there is a long history of looking at ratios (e.g., Bales, 1950) and non-ratio indices (e.g., Bradburn, 1969) relating positive to negative emotions. However, Fredrickson and Losada took matters considerably farther, claiming to have established that their use of a mathematical model drawn from nonlinear dynamics provided theoretical support for the existence of a pair of critical positivity-ratio values (2.9013 and 11.6346) such that individuals whose ratios fall between these values will "flourish," while people whose ratios lie outside this ideal range will "languish." The same article purported to verify this assertion empirically, by demonstrating that among a group of college students, those who were "languishing" had an average positivity ratio of 2.3, while those who were "flourishing" had an average positivity ratio of 3.2.The work of Fredrickson and Losada (2005) has had an extensive influence on the field of positive psychology. This article has been frequently cited, with the Web of Knowledge listing 322 scholarly citations as of April 25, 2013. Fredrickson and Kurtz (2011, pp. 41-42), in a recent review, highlighted this work as providing an "evidence-based guideline" for the claim that a specific value of the positivity ratio acts as a "tipping point beyond which the full impa...