Abstract Circadian rhythms are very important mechanisms which regulate many processes in many organisms. It is believed they are driven via transcription in higher organisms or via phosphorylation (ie., informally, an activation mechanism) in lower organisms.

We are interested in a model of a clock present in Cyanobacteria, which involves the rhythmic phosphorylation of the protein KaiC. The KaiC protein forms complexes of six KaiC joined together (a hexamer). Therefore a single KaiC hexamer can exist in 7 stages of phosphorylation with 0 to all 6 of its single KaiC proteins phosphorylated.

The clock has been shown to persist in vitro in the presence of only KaiA, KaiB, KaiC and ATP. KaiA and KaiB form dimers and are known to form complexes with KaiC hexamers although such polymers do not contain more than a single KaiC hexamer. So we know that a single KaiC hexamer cycles through these states of varying degrees of phosphorylation. The question is can you produce a model in which there is an oscillating concentration/population of phosphorylated KaiC hexamers (expectedly regulated by KaiA and KaiB)? In addition does your model still allow this when there is a non-saturated population of unbound KaiA protein, that is the concentration/population of unbound KaiA dimers is not close to zero?. KaiA appears to stimulate phosphorylation whilst KaiB appears to interfere with this effect.

One possibility is that the KaiC monomers may be in either an active or inactive state, whereby the active state promotes phosphorylation and the inactive state promotes dephosporylation. It is thought that the energetic cost of a single KaiC hexamer containing two monomers in different active-inactive configurations is prohibitively high and therefore the entire KaiC hexamer is either in the active or the inactive state. Therefore a single KaiC hexamer can exist in 14 states corresponding to the 7 degrees of phosphorylation in both the active and inactive states. More information and a model of this explanation can be found in van Zon et 2007[1].

References
[1] van Zon, Lubensky, Altena, ten Wolde. An allosteric model of circadian KaiC phosphorylation. PNAS 104(18) (2007) 7420-7425
See also
[2]Hakuto Kageyama, Takao Kondo and Hideo Iwasaki. Circadian Formation of Clock Protein Complexes by KaiA, KaiB, KaiC, and SasA in Cyanobacteria

Abstract The analysis of crowd dynamics shows emergent patterns of behaviour, which result from the interaction of the many individuals comprising a crowd. In this case the behaviour of a crowd evacuating a prototype building has to be modelled. The initial conditions of the system are described in Appendix C of [1]. This is the user manual of EVACNET4, a tool for the modelling of building egress based on a capacitated network flow transhipment algorithm and used for the generation of optimal building evacuation plans. The case study description includes details of human factors such as how fast people cover a certain distance on average and how many people may pass on average through a standard door in a given time period. The mentioned appendix contains a detail description of a three storey building and assumptions about "standard" human behaviour that can be expected in the described situation.

Being based on an optimisation approach, the tool does not appear flexible enough to model the many "non-optimal" evacuation patterns and different human behaviours that can occur during an evacuation. The question is how can we devise a model of the dynamics of the building egress that is "safe", as it properly models aspects such as egress times and room occupancy, and "expressive", as it allow individual behaviours and non-standard situations - the one that more often recur in emergencies - to be easily embedded in the model and analysis?

You should use the results in [1] as a sanity check for your model. You may take inspiration from [2], about examples of behaviours and situations you may wish to embed in your model for making it more realistic and use it for interesting analysis exploiting your preferred techniques.

References
[1] T. M. Kisko, R. L. Francis and C. R. Nobel. EVACNET4 USER'S GUIDE University of Florida
[2] Gabriel Santos and Benigno Aguirre. A critical review of Emergency evacuation simulation models NIST ws on Building Occupant Movement during Fire Emergency.

Abstract In recent years the economics of information security has been an important topic of research (for a survey see [1], and for a cost-benefit analysis see [2]). In [3] a mathematical model is presented based on the confidentiality and availability trade-off involved in patching. This model uses a method called linear stabilization. Specifically a decision maker minimises a loss function, defined in terms of the confidentiality C(t), Availability A(t) and investment K(t). A solution is defined as a definite integral. Patches are interpreted as shocks to C(t) and A(t) and are assumed to have Poisson-type arrival times and bivariate log-normal intensities. Numerical simulations are generated for different parameter choices. In particular, the paper considers the difference in patching profiles observed for military and financial software systems.

The model could be extended in various directions. For instance, others approaches than numerical simulations could be introduced and strong assumptions (e.g. in the distribution of arrival times) relaxed. Flexibility in the sense of allowing for alternative assumptions could be sought as well. The question is how can we devise a model of the system that clarifies the system under investigation and either:
- Allows for verification, or, at the very least allows for more extensive simulation
- Allows for a less restrictive (or at least an alternative) set of assumptions?
You should use the model in [3] to inform your model.

References
[1] R. Anderson and T. Moore. The economics of information Security. Science, 314:610-613, 2006.
[2] L. Gordon and M. Loeb. Managing Cybersecurity Resources: A cost-benefit Analysis. McGraw Hill, 2006.
[3] Christos Ioannidis, David Pym, and Julian Williams.Information Security Trade-offs and Optimal Patching Policies. European Journal of Operational Research, 216(2):434-444, 16 January 2012. doi:10.1016/j.ejor.2011.05.050.
Pre-print available at http://www.abdn.ac.uk/~csc335/IoannidisPymWilliams-EconPatching.pdf