PubMed · 8289139
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H Aronson. 1993. Correction factor for dissolution profile calculations.. https://doi.org/10.1002/jps.2600821126
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Many small proteins fold fast and without detectable intermediates. This is frequently taken as evidence against the importance of partially folded states, which often transiently accumulate during folding of larger proteins. To get insight into the properties of free energy barriers in protein folding we analyzed experimental data from 23 proteins that were reported to show non-linear activation free-energy relationships. These non-linearities are generally interpreted in terms of broad transition barrier regions with a large number of energetically similar states. Our results argue against the presence of a single broad barrier region. They rather indicate that the non-linearities are caused by sequential folding pathways with consecutive distinct barriers and a few obligatory high-energy intermediates. In contrast to a broad barrier model the sequential model gives a consistent picture of the folding barriers for different variants of the same protein and when folding of a single protein is analyzed under different solvent conditions. The sequential model is also able to explain changes from linear to non-linear free energy relationships and from apparent two-state folding to folding through populated intermediates upon single point mutations or changes in the experimental conditions. These results suggest that the apparent discrepancy between two-state and multi-state folding originates in the relative stability of the intermediates, which argues for the importance of partially folded states in protein folding.
A new empirical equation for the sigmoid pattern of determinate growth, 'the beta growth function', is presented. It calculates weight (w) in dependence of time, using the following three parameters: t(m), the time at which the maximum growth rate is obtained; t(e), the time at the end of growth; and w(max), the maximal value for w, which is achieved at t(e). The beta growth function was compared with four classical (logistic, Richards, Gompertz and Weibull) growth equations, and two expolinear equations. All equations described successfully the sigmoid dynamics of seed filling, plant growth and crop biomass production. However, differences were found in estimating w(max). Features of the beta function are: (1) like the Richards equation it is flexible in describing various asymmetrical sigmoid patterns (its symmetrical form is a cubic polynomial); (2) like the logistic and the Gompertz equations its parameters are numerically stable in statistical estimation; (3) like the Weibull function it predicts zero mass at time zero, but its extension to deal with various initial conditions can be easily obtained; (4) relative to the truncated expolinear equation it provides more reasonable estimates of final quantity and duration of a growth process. In addition, the new function predicts a zero growth rate at both the start and end of a precisely defined growth period. Therefore, it is unique for dealing with determinate growth, and is more suitable than other functions for embedding in process-based crop simulation models to describe the dynamics of organs as sinks to absorb assimilates. Because its parameters correspond to growth traits of interest to crop scientists, the beta growth function is suitable for characterization of environmental and genotypic influences on growth processes. However, it is not suitable for estimating maximum relative growth rate to characterize early growth that is expected to be close to exponential.
We investigate synchronization phenomena in a system of two piecewise-linear-type model neurons with excitatory or inhibitory synaptic couplings. Employing the phase plane analysis and a singular perturbation approach to split the dynamics into slow and fast ones, we construct analytically the Poincaré map of the solution to the piecewise-linear equations. We investigate conditions for the occurrence of synchronized oscillations of in phase as well as of antiphase in terms of parameters representing the strength of the synaptic coupling and the decaying relaxation rate of the synaptic dynamics. We present the results of numerical simulations that agree with our theoretical ones.