This paper is an analytical study of Boolean networks. The motivation is our desire to understand the large, complicated and interconnected pathways which comprise intracellular biochemical signal transduction networks. The simplest possible conceptual model that mimics signal transduction with sigmoidal kinetics is the n-node Boolean network each of whose elements or nodes has the value 0 (off) or 1 (on) at any given time T = 0, 1, 2, …. A Boolean network has 2nstates all of which are either on periodic cycles (including fixed points) or transients leading to cycles. Thus one understands a Boolean network by determining the number and length of its cycles. The problem one must circumvent is the large number of states (2n) since the networks we are interested in have 100 or more elements. Thus we concentrate on developing size n methods rather than the impossible task of enumerating all 2nstates. This is done as follows: the dynamics of the network can be described by n polynomial equations which describe the logical function which determines the interaction at each node. Iterating the equations one step at a time finds all fixed points, period two cycles, period three cycles, etc. This is a general method that can be used to determine the fixed points and moderately large periodic cycles of any size network, but it is not useful in finding the largest cycles in a large network. However, we also show that the network equations can often be reduced to scalar form, which makes the cycle structure much more transparent. The scalar equations method is a true "size n" method and several examples (including nontrivial biochemical systems) are examined.

One way of coping with the complexity of biological systems is to use the simplest possible models which are able to reproduce at least some nontrivial features of reality. Although two value Boolean models have a long history in technology, it is perhaps a little bit surprising that they can also represent important features of living organizms. In this paper, the scalar equation approach to Boolean network models is further developed and then applied to two interesting biological models. In particular, a linear reduced scalar equation is derived from a more rudimentary nonlinear scalar equation. This simpler, but higher order, two term equation gives immediate information about both cycle and transient structure of the network.

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Understanding the abundance and fate of human viral pathogens in wastewater is essential when assessing the public health risks associated with wastewater discharge to the environment. Typically, however, the microbiological monitoring of wastewater is undertaken on an infrequent basis and peak discharge events may be missed leading to the misrepresentation of risk levels. To evaluate diurnal patterns in wastewater viral loading, we undertook 3-day sampling campaigns with bi-hourly sample collection over three seasons at three wastewater treatment plants. Untreated influent was collected at Ganol and secondary-treated effluent was sampled at Llanrwst and Betws-y-Coed (North Wales, UK). Our results confirmed the presence of human adenovirus (AdV), norovirus genotypes I and II (NoVGI and NoVGII) in both influent and effluent samples while sapovirus GI (SaVGI) was only detected in influent water. The AdV titre was high and relatively constant in all samples, whereas the NoVGI, NoVGII and SaVGI showed high concentrations during autumn and winter and low counts during the summer. Diurnal patterns were detected in pH and turbidity for some sampling periods; however, no such changes in viral titres were observed apart from slight fluctuations in the influent samples. Our findings suggest that viral particle number in wastewater is not affected by daily chemical fluctuations. Hence, a grab sample taken at any point during the day may be sufficient to enumerate the viral load of wastewater effluent within an order of magnitude while four samples a day are recommended for testing wastewater influent samples.

The Hill equation is a fundamental expression in chemical ikinetics relating velocity of response to concentration. It is known that the Hill equation is parameter identifiable in the sense that perfect data yield a unique set of defining parameters. However not all sigmoidal curves can be well fit by Hill curves. In particular the lower part of the curve can't be too shallow and the upper part can't be too steep. In this paper an exact mathematical criterion is derived to describe the degree of shallowness allowed.