  How do single cell fate decisions induced by activation of key signaling
proteins above threshold concentrations within a time interval are affected by
stochastic fluctuations in biochemical reactions? We address this question
using minimal models of stochastic chemical reactions commonly found in cell
signaling and gene regulatory systems. Employing exact solutions and
semi-analytical methods we calculate distributions of the maximum value ($N$)
of activated species concentrations ($P_{max}(N)$) and the time ($t$) taken to
reach the maximum value ($P_{max}(t)$) within a time window in the minimal
models. We find, the presence of positive feedback interactions make
$P_{max}(N)$ more spread out with a higher "peakedness" in $P_{max}(t)$. Thus
positive feedback interactions may help single cells to respond sensitively to
a stimulus when cell decision processes require upregulation of activated forms
of key proteins to a threshold number within a time window.
