Propagation of fluctuations in biochemical systems, II: Nonlinear chains.

We consider biochemical reaction chains and investigate how random external fluctuations, as characterised by variance and coefficient of variation, propagate down the chains. We perform such a study under the assumption that the number of molecules is high enough so that the behaviour of the concentrations of the system is well approximated by differential equations. We conclude that the variances and coefficients of variation of the fluxes will decrease as one moves down the chain and, through an example, show that there is no corresponding result for the variances of the concentrations of the chemical species. We also prove that the fluctuations of the fluxes as characterised by their time averages decrease down reaction chains. The results presented give insight into how biochemical reaction systems are buffered against external perturbations solely by their underlying graphical structure and point out the benefits of studying the out-of-equilibrium dynamics of systems.

Teaching multiplication facts to students with learning problems.

The present study investigated the effectiveness of a constant time delay procedure in teaching multiplication facts to fifth- and sixth-grade students identified as learning disabled, behavior disordered, or educable mentally handicapped. Four students were taught oral responses to 30 multiplication facts using a constant 5-second (s) delay procedure. Data on the number of minutes of direct instruction time, as well as the number and percent of errors, trials, and sessions to criterion, were collected. Results indicated that the near-errorless learning procedure was effective in teaching multiplication facts to students placed in special education.